%------------------------------------------------------------------------------
% File     : ITP249^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Bounds 00615_028958
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0071_VEBT_Bounds_00615_028958 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11243 (6005 unt; 999 typ;   0 def)
%            Number of atoms       : 27151 (12372 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 114082 (2654   ~; 480   |;1888   &;99500   @)
%                                         (   0 <=>;9560  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   97 (  96 usr)
%            Number of type conns  : 4097 (4097   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  906 ( 903 usr;  55 con; 0-8 aty)
%            Number of variables   : 25443 (2243   ^;22395   !; 805   ?;25443   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 02:28:44.371
%------------------------------------------------------------------------------
% Could-be-implicit typings (96)
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__VEBT____Definitions__OVEBT,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Rat__Orat,type,
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thf(ty_n_t__Num__Onum,type,
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thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (903)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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thf(sy_c_Binomial_Obinomial,type,
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thf(sy_c_Bit__Operations_Oand__int__rel,type,
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thf(sy_c_Bit__Operations_Oand__not__num,type,
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thf(sy_c_Bit__Operations_Oconcat__bit,type,
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thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
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thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
    the_real: ( real > $o ) > real ).

thf(sy_c_If_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).

thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(sy_c_If_001t__Complex__Ocomplex,type,
    if_complex: $o > complex > complex > complex ).

thf(sy_c_If_001t__Extended____Nat__Oenat,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Num__Onum,type,
    if_num: $o > num > num > num ).

thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Onat,type,
    nat2: int > nat ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: int > int ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
    ring_1_of_int_rat: int > rat ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
    ring_1_of_int_real: int > real ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
    sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    sup_sup_set_set_o: set_set_o > set_set_o > set_set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( nat > real ) > filter_nat > $o ).

thf(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: list_int > list_int > list_int ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: list_int > $o ).

thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

thf(sy_c_List_Olist__update_001_Eo,type,
    list_update_o: list_o > nat > $o > list_o ).

thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
    list_update_complex: list_complex > nat > complex > list_complex ).

thf(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: list_int > nat > int > list_int ).

thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: list_real > nat > real > list_real ).

thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
    list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).

thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > nat > $o ).

thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > nat > code_integer ).

thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
    nth_complex: list_complex > nat > complex ).

thf(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > nat > int ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Num__Onum,type,
    nth_num: list_num > nat > num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).

thf(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > nat > real ).

thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
    product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
    product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
    product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
    product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
    product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: list_num > list_num > list_P3744719386663036955um_num ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: nat > $o > list_o ).

thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
    replicate_complex: nat > complex > list_complex ).

thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: nat > int > list_int ).

thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: nat > nat > list_nat ).

thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: nat > real > list_real ).

thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: nat > set_nat > list_set_nat ).

thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Oupto,type,
    upto: int > int > list_int ).

thf(sy_c_List_Oupto__aux,type,
    upto_aux: int > int > list_int > list_int ).

thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

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

thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

thf(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

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_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
    size_size_list_num: list_num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
    size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

thf(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

thf(sy_c_NthRoot_Oroot,type,
    root: nat > real > real ).

thf(sy_c_NthRoot_Osqrt,type,
    sqrt: real > real ).

thf(sy_c_Num_OBitM,type,
    bitM: num > num ).

thf(sy_c_Num_Oinc,type,
    inc: num > num ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > num > int ).

thf(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

thf(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: num > int ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

thf(sy_c_Num_Opow,type,
    pow: num > num > num ).

thf(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

thf(sy_c_Num_Osqr,type,
    sqr: num > num ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: int > ( num > int ) > option_num > int ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: option_num > num ).

thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

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__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
    ord_less_set_o: set_o > set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > set_real > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $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__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_eq_set_num: set_num > set_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > int > int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > num > num ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: rat > rat > rat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > real > real ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: ( nat > real ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: ( nat > real ) > $o ).

thf(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_mono_real_real: ( real > real ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > nat > complex ).

thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > nat > int ).

thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > nat > nat ).

thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > nat > rat ).

thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: real > nat > real ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Rat_OFract,type,
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thf(sy_c_Rat_OFrct,type,
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thf(sy_c_Rat_Onormalize,type,
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thf(sy_c_Rat_Oof__int,type,
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thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: ( nat > real ) > real ).

thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osums_001t__Int__Oint,type,
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thf(sy_c_Series_Osums_001t__Nat__Onat,type,
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thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Num__Onum,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

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

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

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

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

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: rat > set_rat > set_rat ).

thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: num > num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: rat > rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001_Eo,type,
    set_or6416164934427428222Than_o: $o > set_o ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
    set_ord_lessThan_o: $o > set_o ).

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

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

thf(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

thf(sy_c_String_Ochar_OChar,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

thf(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: char > nat ).

thf(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: nat > char ).

thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(sy_c_Topological__Spaces_Ogenerate__topology_001_Eo,type,
    topolo4667128019001906403logy_o: set_set_o > set_o > $o ).

thf(sy_c_Topological__Spaces_Ogenerate__topology_001t__Nat__Onat,type,
    topolo1613498594424996677gy_nat: set_set_nat > set_nat > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Code____Numeral__Ointeger,type,
    topolo2919662092509805066nteger: ( nat > code_integer ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
    topolo4899668324122417113eq_int: ( nat > int ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: ( nat > num ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: ( nat > set_nat ) > $o ).

thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001_Eo,type,
    topolo9180104560040979295open_o: set_o > $o ).

thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
    topolo4110288021797289639omplex: set_complex > $o ).

thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
    topolo4328251076210115529en_nat: set_nat > $o ).

thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
    topolo4860482606490270245n_real: set_real > $o ).

thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: real > set_real > filter_real ).

thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: ( nat > complex ) > $o ).

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

thf(sy_c_Transcendental_Oarccos,type,
    arccos: real > real ).

thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: real > real ).

thf(sy_c_Transcendental_Oarcsin,type,
    arcsin: real > real ).

thf(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: real > real ).

thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: real > real ).

thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: complex > complex ).

thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: real > real ).

thf(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: real > real ).

thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: real > real ).

thf(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
    diffs_Code_integer: ( nat > code_integer ) > nat > code_integer ).

thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: ( nat > complex ) > nat > complex ).

thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: ( nat > int ) > nat > int ).

thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: ( nat > rat ) > nat > rat ).

thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: ( nat > real ) > nat > real ).

thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: complex > complex ).

thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: real > real ).

thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: real > real ).

thf(sy_c_Transcendental_Olog,type,
    log: real > real > real ).

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > real > real ).

thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: complex > complex ).

thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: real > real ).

thf(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: real > real ).

thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: complex > complex ).

thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: real > real ).

thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: complex > complex ).

thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: real > real ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: nat > nat > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > set_complex > $o ).

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

thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: list_o > set_list_o > $o ).

thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: list_int > set_list_int > $o ).

thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Num__Onum,type,
    member_num: num > set_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10205)
thf(fact_0_True,axiom,
    ord_less_nat @ xa @ mi ).

% True
thf(fact_1_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] :
          ( X
         != ( plus_plus_nat @ N @ N ) )
     => ~ ! [N: nat] :
            ( X
           != ( plus_plus_nat @ N @ ( suc @ N ) ) ) ) ).

% even_odd_cases
thf(fact_2_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_3__C4_Ohyps_C_I7_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "4.hyps"(7)
thf(fact_4__C4_Ohyps_C_I8_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "4.hyps"(8)
thf(fact_5_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_6_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_7_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_8_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_9_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_10_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_11_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_12_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_13_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_14_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_15_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_16_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_17_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_18_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_19_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_20_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_21_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_22_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_23_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_24_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_25_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_26__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_A_Ideg_A_N_A2_J_J_J_AtreeList_Asummary_J_Ax_A_061_A19_A_L_A_Ilet_Axn_A_061_Aif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_059_Aminn_A_061_Aif_Ax_A_060_Ami_Athen_Ax_Aelse_Ami_059_Al_A_061_Alow_Axn_A_ISuc_A_ISuc_A_Ideg_A_N_A2_J_J_Adiv_A2_J_059_Ah_A_061_Ahigh_Axn_A_ISuc_A_ISuc_A_Ideg_A_N_A2_J_J_Adiv_A2_J_Ain_Aif_Ah_A_060_Alength_AtreeList_A_092_060and_062_A_092_060not_062_A_Ix_A_061_Ami_A_092_060or_062_Ax_A_061_Ama_J_Athen_AT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_A_ItreeList_A_B_Ah_J_Al_A_L_AT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_A_ItreeList_A_B_Ah_J_A_L_A_Iif_AminNull_A_ItreeList_A_B_Ah_J_Athen_AT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Asummary_Ah_Aelse_A1_J_Aelse_A1_J_092_060close_062,axiom,
    ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ treeList @ summary ) @ xa )
    = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
      @ ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
          & ~ ( ( xa = mi )
              | ( xa = ma ) ) )
        @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ ( minus_minus_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ) ).

% \<open>T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t (Node (Some (mi, ma)) (Suc (Suc (deg - 2))) treeList summary) x = 19 + (let xn = if x < mi then mi else x; minn = if x < mi then x else mi; l = low xn (Suc (Suc (deg - 2)) div 2); h = high xn (Suc (Suc (deg - 2)) div 2) in if h < length treeList \<and> \<not> (x = mi \<or> x = ma) then T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t (treeList ! h) l + T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l (treeList ! h) + (if minNull (treeList ! h) then T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t summary h else 1) else 1)\<close>
thf(fact_27_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera6690914467698888265omplex @ N2 )
        = one_one_complex )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_28_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_real @ N2 )
        = one_one_real )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_29_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_30_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_31_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_32_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_33_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_34_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_35_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_36_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_37_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_38_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_39_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_40_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_41_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_42_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X2: nat] :
          ( ( member_nat @ X2 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).

% max_in_set_def
thf(fact_43_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X2: nat] :
          ( ( member_nat @ X2 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% min_in_set_def
thf(fact_44_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_45_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_46_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_47_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_48_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_49_semiring__norm_I87_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(87)
thf(fact_50_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_51_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_52_semiring__norm_I90_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(90)
thf(fact_53_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_54_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X2: nat,N3: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% high_def
thf(fact_55__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% \<open>2 \<le> deg\<close>
thf(fact_56_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_57_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_58_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_59_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_60_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_61_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_62_semiring__norm_I83_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit0 @ N2 ) ) ).

% semiring_norm(83)
thf(fact_63_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_64_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_65_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_66_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_67_semiring__norm_I88_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N2 ) ) ).

% semiring_norm(88)
thf(fact_68_semiring__norm_I89_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N2 ) ) ).

% semiring_norm(89)
thf(fact_69_semiring__norm_I84_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit1 @ N2 ) ) ).

% semiring_norm(84)
thf(fact_70_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_71_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_72_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_73_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_86_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_87_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_88_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_89_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_90_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_91_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_92_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_93_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_94_semiring__norm_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(6)
thf(fact_95__C4_Ohyps_C_I2_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "4.hyps"(2)
thf(fact_96_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(78)
thf(fact_97_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_98_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(80)
thf(fact_99_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_100_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_101_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_102_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_103_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_104_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_105_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_106_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_107_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_108_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_109_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_110_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_111_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_112_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_113_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_114_semiring__norm_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(7)
thf(fact_115_semiring__norm_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(9)
thf(fact_116_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_117_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_118_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_119_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_120_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_121_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(81)
thf(fact_122_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_123_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_124_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_125_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_126_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_127_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_128_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_129_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_130_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_131_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_132_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_133_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_134_semiring__norm_I10_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_135_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_136_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_137_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_138_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_139_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_140_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_141_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_142_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_143_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_144_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_145_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_146_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_147_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_148_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_149_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_150_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_151_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_152_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_153_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_154_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_155_Suc__le__D,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 )
     => ? [M3: nat] :
          ( M2
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_156_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_157_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_158_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_159_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_160_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_161_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_162_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_163_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_164_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_165_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_166_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_167_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_168_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_169_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_170_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_171_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_172_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_173_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_174_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
             => ( P @ M4 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_175_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_176_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_177_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ M @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_178_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
           => ( R @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_179_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_180_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_181_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_182_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_183_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_184_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_185_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_186_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ ( suc @ N ) )
                 => ( P @ N ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_187_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ N )
                 => ( P @ ( suc @ N ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_188_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_189_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_190_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_191_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_192_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_193_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_194_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_195_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_196_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_197_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_198_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_199_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_200_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_201_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_202_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_203_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_204_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
          | ( M5 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_205_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_206_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
          & ( M5 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_207_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_208_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_209_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_210_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_211_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_212_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N: nat] :
          ( L
          = ( plus_plus_nat @ K @ N ) ) ) ).

% le_Suc_ex
thf(fact_213_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_214_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_215_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_216_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_217_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_218_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_219_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_220_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_221_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_222_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_223_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_224_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_225_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_226_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_227_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_228_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_229_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_230_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_231_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_232_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_233_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_234_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_235_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_236_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_237_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_238_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_nat @ M3 @ N )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_239_add__diff__add,axiom,
    ! [A: complex,C: complex,B: complex,D: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) )
      = ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ ( minus_minus_complex @ C @ D ) ) ) ).

% add_diff_add
thf(fact_240_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_241_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_242_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_243_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_244_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_245_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_246_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_247_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I2 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_248_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_249_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_250_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_251_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N2 @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_252_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N2 )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_253_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_254_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_255_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N2 )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_256_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_257_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_258_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_259_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_260_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_261_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_262_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_263_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_264_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_265_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_266_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_267_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_268_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_269_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_270_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ one_one_nat ) ).

% minNull_bound
thf(fact_271_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_272_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_273_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_274_is__num__normalize_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_275_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_276_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( P @ N )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
              & ~ ( P @ M4 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_277_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
             => ( P @ M4 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_278_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_279_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_280_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_281_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_282_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_283_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_284_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_285_size__neq__size__imp__neq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_VEBT_VEBT @ X )
       != ( size_size_VEBT_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_286_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_287_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_288_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_289_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_290_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_291_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_292_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_293_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_294_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_295_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_296_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_297_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_298_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_299_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_300_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_301_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_302_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_303_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_304_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q2: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_305_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_306_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_307_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_308_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_309_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_310_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_311_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_312_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_313_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_314_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_315_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_316_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_317_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_318_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_319_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_320_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_321_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_322_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_323_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_324_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_325_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_326_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_327_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_328_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_329_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_330_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_331_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_332_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_333_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_334_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_335_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_336_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_337_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_338_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_339_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_340_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_341_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_342_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_343_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_344_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_345_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_346_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X32: num] :
              ( Y
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_347_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_348_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_349_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_350_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_351_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_352_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_353_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_354_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_355_field__sum__of__halves,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_356_field__sum__of__halves,axiom,
    ! [X: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_357_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_358_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_359_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_360_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_361_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_362_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_363_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_364_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_365_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_366_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_367_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_368_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_369_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_370_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_371_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_372_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_373_power__inject__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_374_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_375_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_376_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_377_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_378_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_379_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_380_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_381_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_382_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_383_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_384_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_385__C4_Ohyps_C_I3_J,axiom,
    m = na ).

% "4.hyps"(3)
thf(fact_386__C4_Ohyps_C_I5_J,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X4 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).

% "4.hyps"(5)
thf(fact_387__C4_Ohyps_C_I4_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "4.hyps"(4)
thf(fact_388_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_389_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_390_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_391_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_392_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_393_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_394_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_395_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ one_one_real @ N2 )
      = one_one_real ) ).

% power_one
thf(fact_396_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ one_one_complex @ N2 )
      = one_one_complex ) ).

% power_one
thf(fact_397_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_398_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_399_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_400_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_401_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_402_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(71)
thf(fact_403_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_404_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(73)
thf(fact_405_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_406_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_407_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_408_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_409_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_410_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(72)
thf(fact_411_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_412_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(79)
thf(fact_413_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(74)
thf(fact_414__C4_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X5: nat] :
              ( ( ( ( vEBT_VEBT_high @ X5 @ na )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X5 )
                & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).

% "4.hyps"(9)
thf(fact_415__C4_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).

% "4.hyps"(6)
thf(fact_416__C4_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "4.hyps"(1)
thf(fact_417_two__realpow__ge__one,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% two_realpow_ge_one
thf(fact_418_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_419_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_420_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_421_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_422_power__divide,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).

% power_divide
thf(fact_423_power__divide,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
      = ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).

% power_divide
thf(fact_424_power__divide,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N2 )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).

% power_divide
thf(fact_425_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_426_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_427_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_428_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_429_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_430_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_431_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_432_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_433_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_434_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_435_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_436_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_437_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N2: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
         => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_438_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_439_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_440_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_441_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_442_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_443_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_444_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_445_one__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_446_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_447_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_448_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_449_power__one__over,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_450_power__one__over,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_451_power__one__over,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_452_power__gt1,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_453_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_454_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_455_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_456_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_457_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_458_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_459_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_460_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_461_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_462_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_463_power__strict__increasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_464_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_465_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_466_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_467_power__increasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_468_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_469_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_470_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_471_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_472_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_473_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_474_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_475_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_476_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_477_power2__commute,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_478_power2__commute,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_479_power2__commute,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_480_power2__commute,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_481_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_482_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_483_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_484_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_485_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_486_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N3: nat,TreeList2: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ N3 ) ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) ) ) ).

% in_children_def
thf(fact_487_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_488_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_489_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_490_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_491_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_492_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_493_low__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = X ) ) ).

% low_inv
thf(fact_494_high__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = Y ) ) ).

% high_inv
thf(fact_495_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_496_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% height_compose_summary
thf(fact_497_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_498_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_499_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_500_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_501_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_502_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N2: nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_503_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N2: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_504_inthall,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,N2: nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( P @ ( nth_set_nat @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_505_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_506_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_507_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N2: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_508_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N2: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_509_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_compose_child
thf(fact_510_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_511_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_512_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_513_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_514_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_515_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_516_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_517_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_518_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_519_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_520_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_521_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_522_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_523_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_524_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% two_powr_height_bound_deg
thf(fact_525_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_526_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_527_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_528_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_529_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_530__C4_OIH_C_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X5 @ na )
        & ! [Xa: nat] : ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ X5 @ Xa ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ X5 ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% "4.IH"(1)
thf(fact_531_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_532_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_533_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_534_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_535_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_536_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_537_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_538_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_539_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_540_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_541_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_542_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_543_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_544_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_545_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_546_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_547_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_548_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_549_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_550_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_551_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_552_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ M @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_553__C4_OIH_C_I2_J,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ summary @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ summary ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% "4.IH"(2)
thf(fact_554_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_555_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_556_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_557_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_558_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_559_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_560_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_561_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_562_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_563_power__add__numeral,axiom,
    ! [A: complex,M: num,N2: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_564_power__add__numeral,axiom,
    ! [A: real,M: num,N2: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_565_power__add__numeral,axiom,
    ! [A: rat,M: num,N2: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_566_power__add__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_567_power__add__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_568_power__add__numeral2,axiom,
    ! [A: complex,M: num,N2: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_569_power__add__numeral2,axiom,
    ! [A: real,M: num,N2: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_570_power__add__numeral2,axiom,
    ! [A: rat,M: num,N2: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_571_power__add__numeral2,axiom,
    ! [A: nat,M: num,N2: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_572_power__add__numeral2,axiom,
    ! [A: int,M: num,N2: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_573_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ X @ Z3 ) )
           => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).

% succ_member
thf(fact_574_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ Z3 @ X ) )
           => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).

% pred_member
thf(fact_575_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X5: real] : ( member_real @ X5 @ S3 )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S3 )
           => ( ord_less_eq_real @ X3 @ Z4 ) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ord_less_eq_real @ X5 @ Y3 ) )
            & ! [Z4: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S3 )
                   => ( ord_less_eq_real @ X3 @ Z4 ) )
               => ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).

% complete_real
thf(fact_576_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N ) ) ) ).

% real_arch_pow
thf(fact_577_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_real @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_578_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_579_combine__common__factor,axiom,
    ! [A: complex,E: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_580_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_581_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_582_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_583_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_584_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% distrib_right
thf(fact_585_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_586_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_587_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_588_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_589_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% distrib_left
thf(fact_590_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_591_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_592_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_593_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_594_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_595_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_596_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_597_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_598_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_599_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_600_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_601_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_602_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_603_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_604_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_605_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_606_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_607_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_608_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_609_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_610_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_611_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_612_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_613_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_614_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_615_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_616_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_617_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_618_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_619_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_620_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_621_power__commutes,axiom,
    ! [A: complex,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_622_power__commutes,axiom,
    ! [A: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_623_power__commutes,axiom,
    ! [A: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_624_power__commutes,axiom,
    ! [A: nat,N2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_625_power__commutes,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_626_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_627_power__mult__distrib,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_628_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_629_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_630_power__mult__distrib,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_631_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_632_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_633_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_634_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_635_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_636_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_637_power__mult,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_638_power__mult,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_639_power__mult,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_640_power__mult,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_641_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_642_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_643_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_644_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_645_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_646_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_647_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_648_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_649_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_650_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_651_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_652_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_653_div__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q3 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q3 ) ) ).

% div_mult2_eq
thf(fact_654_lambda__one,axiom,
    ( ( ^ [X2: complex] : X2 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_655_lambda__one,axiom,
    ( ( ^ [X2: real] : X2 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_656_lambda__one,axiom,
    ( ( ^ [X2: rat] : X2 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_657_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_658_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_659_less__1__mult,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N2 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_660_less__1__mult,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_661_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_662_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_663_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_664_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_665_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_666_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_667_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_668_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_669_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_670_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_671_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_672_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_673_mult__diff__mult,axiom,
    ! [X: complex,Y: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_674_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_675_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_676_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_677_square__diff__square__factored,axiom,
    ! [X: complex,Y: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
      = ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_678_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_679_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_680_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_681_eq__add__iff2,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_682_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_683_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_684_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_685_eq__add__iff1,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_686_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_687_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_688_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_689_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_690_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_691_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_692_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_693_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_694_power__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_695_power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_696_power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_697_power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_698_power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_699_power__Suc2,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_700_power__Suc2,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_701_power__Suc2,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_702_power__Suc2,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_703_power__Suc2,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_704_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_705_power__add,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_add
thf(fact_706_power__add,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_add
thf(fact_707_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_add
thf(fact_708_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_709_power__add,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_710_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_711_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_712_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_713_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_714_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_715_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_716_power__odd__eq,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_717_power__odd__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_718_power__odd__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_719_power__odd__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_720_power__odd__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_721_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_node
thf(fact_722_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_723_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_724_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_725_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_726_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_727_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_728_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_729_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_730_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_731_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_732_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_733_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_734_power__less__power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_735_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_736_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_737_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_738_power__gt1__lemma,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_739_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_740_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_741_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_742_square__diff__one__factored,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_743_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_744_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_745_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_746_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_747_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_748_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_749_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_750_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_751_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_752_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_753_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_754_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_755_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_756_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_757_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_758_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_759_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_760_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_761_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_762_left__add__twice,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_763_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_764_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_765_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_766_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_767_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_768_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_769_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_770_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_771_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_772_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_773_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_774_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_775_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_776_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_777_power2__eq__square,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A @ A ) ) ).

% power2_eq_square
thf(fact_778_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_779_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_780_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_781_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_782_power4__eq__xxxx,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_783_power4__eq__xxxx,axiom,
    ! [X: real] :
      ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_784_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_785_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_786_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_787_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_788_power__even__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_789_power__even__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_790_power__even__eq,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_791_power__even__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_792_power3__eq__cube,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_793_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_794_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_795_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_796_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_797_div__nat__eqI,axiom,
    ! [N2: nat,Q3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q3 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_798_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L ) @ one_one_nat ) ) ) ).

% insertsimp'
thf(fact_799_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_800_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_801_insersimp,axiom,
    ! [T: vEBT_VEBT,N2: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_802_insertsimp,axiom,
    ! [T: vEBT_VEBT,N2: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insertsimp
thf(fact_803_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_804_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_805_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_806_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_807_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_808_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_809_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_810_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_811_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_812_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_813_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X4 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_814_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X4 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_815_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_816_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_817_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_818_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_819_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_820_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_821_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_822_add__diff__cancel__right_H,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_823_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_824_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_825_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_826_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_827_add__diff__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
      = ( minus_minus_complex @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_828_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_829_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_830_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_831_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_832_add__diff__cancel__left_H,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_833_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_834_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_835_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_836_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_837_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_838_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_839_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_840_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_841_add__left__cancel,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = ( plus_plus_complex @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_842_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_843_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_844_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_845_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_846_add__right__cancel,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = ( plus_plus_complex @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_847_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_848_real__divide__square__eq,axiom,
    ! [R2: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
      = ( divide_divide_real @ A @ R2 ) ) ).

% real_divide_square_eq
thf(fact_849_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_850_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_851_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_852_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_853_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_854_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_855_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_856_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_857_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_858_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_859_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_860_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_861_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_862_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_863_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_864_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_865_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_866_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_867_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_868_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_869_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_870_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_871_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_872_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_873_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_874_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_875_add__diff__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_876_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_877_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_878_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_879_diff__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_880_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_881_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_882_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_883_add__diff__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
      = ( minus_minus_complex @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_884_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_885_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_886_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_887_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_888_semiring__norm_I13_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% semiring_norm(13)
thf(fact_889_semiring__norm_I12_J,axiom,
    ! [N2: num] :
      ( ( times_times_num @ one @ N2 )
      = N2 ) ).

% semiring_norm(12)
thf(fact_890_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_891_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_892_num__double,axiom,
    ! [N2: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_893_power__mult__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_894_power__mult__numeral,axiom,
    ! [A: real,M: num,N2: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_895_power__mult__numeral,axiom,
    ! [A: complex,M: num,N2: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_896_power__mult__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_897_semiring__norm_I15_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).

% semiring_norm(15)
thf(fact_898_semiring__norm_I14_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).

% semiring_norm(14)
thf(fact_899_semiring__norm_I16_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_900_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N2: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M4: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M4 @ N )
             => ( P @ M4 ) )
         => ( P @ N ) )
     => ( P @ N2 ) ) ).

% enat_less_induct
thf(fact_901_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_902_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_903_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_904_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_905_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_906_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_907_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_908_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_909_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_910_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_911_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_912_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_913_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_914_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_915_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_916_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_917_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_918_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_919_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_920_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_921_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_922_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_923_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_924_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_925_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_926_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_927_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_928_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_929_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_930_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_931_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_932_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_933_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_934_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_935_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_936_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_937_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_938_group__cancel_Oadd1,axiom,
    ! [A2: complex,K: complex,A: complex,B: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( plus_plus_complex @ A2 @ B )
        = ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_939_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B3 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_940_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_941_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_942_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_943_group__cancel_Oadd2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3
        = ( plus_plus_complex @ K @ B ) )
     => ( ( plus_plus_complex @ A @ B3 )
        = ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_944_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_945_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_946_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_947_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_948_add_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).

% add.assoc
thf(fact_949_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_950_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_951_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_952_add_Oleft__cancel,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = ( plus_plus_complex @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_953_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_954_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_955_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_956_add_Oright__cancel,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = ( plus_plus_complex @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_957_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_958_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_959_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_960_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_961_add_Ocommute,axiom,
    ( plus_plus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_962_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_963_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_964_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_965_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_966_add_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( plus_plus_complex @ B @ ( plus_plus_complex @ A @ C ) )
      = ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).

% add.left_commute
thf(fact_967_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_968_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_969_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_970_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_971_add__left__imp__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = ( plus_plus_complex @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_972_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_973_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_974_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_975_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_976_add__right__imp__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = ( plus_plus_complex @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_977_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_978_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_979_diff__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 ) ) ).

% diff_right_commute
thf(fact_980_diff__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 ) ) ).

% diff_right_commute
thf(fact_981_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_982_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_983_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_984_four__x__squared,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_985_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_986_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_987_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_988_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_989_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_990_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_991_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_992_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_993_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_994_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_995_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_996_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_997_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_998_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_999_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1000_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_1001_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_1002_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1003_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1004_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_1005_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_1006_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1007_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1008_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_1009_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_1010_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_1011_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1012_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1013_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_1014_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_1015_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_1016_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1017_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1018_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_1019_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_1020_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1021_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1022_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_1023_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_1024_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1025_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1026_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_1027_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_1028_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1029_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1030_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_1031_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_1032_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1033_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1034_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_1035_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_1036_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1037_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1038_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_1039_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_1040_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1041_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1042_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_1043_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_1044_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1045_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1046_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_1047_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_1048_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1049_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1050_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_1051_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_1052_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1053_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1054_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_1055_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_1056_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1057_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1058_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_1059_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1060_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1061_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_1062_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1063_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1064_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_1065_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1066_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1067_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_1068_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1069_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1070_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_1071_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1072_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1073_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_1074_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1075_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1076_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_1077_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1078_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1079_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_1080_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1081_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1082_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1083_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_1084_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_1085_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1086_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1087_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1088_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1089_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1090_group__cancel_Osub1,axiom,
    ! [A2: complex,K: complex,A: complex,B: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( minus_minus_complex @ A2 @ B )
        = ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1091_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1092_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1093_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_1094_diff__eq__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( ( minus_minus_complex @ A @ B )
        = C )
      = ( A
        = ( plus_plus_complex @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1095_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1096_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1097_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_1098_eq__diff__eq,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( A
        = ( minus_minus_complex @ C @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1099_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1100_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1101_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_1102_add__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1103_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1104_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1105_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_1106_diff__diff__eq2,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( minus_minus_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1107_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1108_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1109_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_1110_diff__add__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1111_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1112_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1113_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_1114_diff__add__eq__diff__diff__swap,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1115_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1116_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1117_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_1118_add__implies__diff,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( ( plus_plus_complex @ C @ B )
        = A )
     => ( C
        = ( minus_minus_complex @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1119_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1120_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1121_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_1122_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_1123_diff__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1124_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1125_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1126_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_1127_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_1128_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1129_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1130_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_1131_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_1132_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1133_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1134_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_1135_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_1136_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1137_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1138_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_1139_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_1140_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1141_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1142_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_1143_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_1144_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1145_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1146_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_1147_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1148_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1149_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_1150_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1161_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1162_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_1163_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1164_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1165_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_1166_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1167_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1168_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1169_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1170_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1171_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_1172_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1173_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1174_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1175_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1176_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1177_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1178_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_1179_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_1180_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_1181_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1182_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1183_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1184_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_1185_nth__mem,axiom,
    ! [N2: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N2 ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1186_nth__mem,axiom,
    ! [N2: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N2 ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1187_nth__mem,axiom,
    ! [N2: nat,Xs2: list_set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs2 @ N2 ) @ ( set_set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1188_nth__mem,axiom,
    ! [N2: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N2 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1189_nth__mem,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1190_nth__mem,axiom,
    ! [N2: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N2 ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1191_nth__mem,axiom,
    ! [N2: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N2 ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1192_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1193_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1194_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_o @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1195_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_int @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1196_in__set__conv__nth,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1197_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1198_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1199_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1200_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1201_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1202_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1203_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I2 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1204_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1205_all__nth__imp__all__set,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,X: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
         => ( P @ ( nth_set_nat @ Xs2 @ I2 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1206_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1207_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1208_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1209_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1210_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X2: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
          | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).

% both_member_options_def
thf(fact_1211_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1212_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N ) ) )
       => ( ! [N: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N ) ) )
         => ( ! [M3: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
           => ( ! [M3: num,N: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N ) ) )
             => ( ! [M3: num,N: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N ) ) )
               => ( ! [M3: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
                 => ( ! [M3: num,N: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N ) ) )
                   => ~ ! [M3: num,N: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_1213_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1214_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1215_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1216_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1217_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B3 )
      = ( ! [X2: complex] :
            ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1218_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B3 )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1219_subset__code_I1_J,axiom,
    ! [Xs2: list_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ B3 )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
           => ( member_set_nat @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1220_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B3 )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1221_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B3 )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1222_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B3 )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1223_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1224_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1225_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1226_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1227_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1228_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1229_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1230_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1231_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1232_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1233_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1234_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1235_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1236_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1237_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1238_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1239_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1240_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1241_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1242_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1243_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1244_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1245_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1246_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1247_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z5: list_nat] : Y5 = Z5 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1248_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : Y5 = Z5 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1249_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z5: list_o] : Y5 = Z5 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I3 )
                = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1250_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z5: list_int] : Y5 = Z5 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I3 )
                = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1251_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: nat] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1252_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: vEBT_VEBT] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1253_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: $o] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1254_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: int] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1255_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1256_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1257_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1258_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I2 )
              = ( nth_int @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1259_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_1260_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_1261_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_1262_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_1263_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1264_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1265_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1266_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1267_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( N2
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_1268_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_1269_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X2: nat,N3: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% low_def
thf(fact_1270_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X2: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ X2 @ Y2 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ X2 @ Z3 )
               => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_1271_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X2: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ Y2 @ X2 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ Z3 @ X2 )
               => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_1272_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_1273_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B3 )
         => ( ( A2 = B3 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_1274_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_1275_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_1276_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_1277_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_1278_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ X5 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_1279_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ A @ X5 ) ) ) ) ).

% succ_none_empty
thf(fact_1280_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1281_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1282_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1283_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_1284_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_1285_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_1286_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_1287_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_1288_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_1289_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1290_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1291_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1292_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1293_minus__mod__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mod_self2
thf(fact_1294_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_1295_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_1296_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_1297_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_1298_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_1299_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_1300_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_1301_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_1302_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_1303_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_1304_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_1305_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_1306_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_1307_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_1308_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_1309_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_1310_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_1311_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_1312_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_1313_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_1314_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_1315_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_1316_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_1317_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_1318_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_1319_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_1320_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_1321_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_1322_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1323_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1324_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1325_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1326_mod__mult__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1327_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1328_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1329_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1330_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1331_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1332_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1333_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1334_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1335_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1336_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1337_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1338_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1339_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1340_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_1341_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_1342_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_1343_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1344_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1345_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1346_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1347_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1348_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1349_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1350_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1351_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1352_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1353_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1354_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1355_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1356_mod__diff__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1357_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1358_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_1359_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_1360_power__mod,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N2 ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_1361_power__mod,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N2 ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_1362_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N2 ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_1363_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_1364_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_1365_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_1366_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1367_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1368_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1369_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1370_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1371_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1372_finite__lists__length__eq,axiom,
    ! [A2: set_o,N2: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ( size_size_list_o @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1373_finite__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ( size_size_list_int @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1374_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ( size_size_list_nat @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1375_finite__lists__length__le,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1376_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1377_finite__lists__length__le,axiom,
    ! [A2: set_o,N2: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1378_finite__lists__length__le,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1379_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1380_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1381_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1382_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1383_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1384_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1385_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1386_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D3: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1387_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1388_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1389_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1390_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1391_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P2: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P2 )
       => ( ( ord_less_nat @ M @ P2 )
         => ( ! [N: nat] :
                ( ( ord_less_nat @ N @ P2 )
               => ( ( P @ N )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N ) @ P2 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_1392_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_1393_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_1394_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N3 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_1395_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_1396_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_1397_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).

% vebt_member.simps(2)
thf(fact_1398_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_1399_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1400_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1401_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1402_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1403_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1404_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1405_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1406_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1407_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1408_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1409_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1410_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1411_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_1412_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_1413_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_1414_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1415_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1416_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1417_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1418_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1419_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1420_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1421_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1422_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1423_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1424_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1425_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1426_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1427_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1428_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1429_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_1430_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_1431_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_1432_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1433_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1434_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1435_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_1436_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_1437_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_1438_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_1439_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_1440_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_1441_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_1442_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_1443_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_1444_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_1445_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_1446_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_1447_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_1448_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_1449_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_1450_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q2: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_1451_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S2: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_1452_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_1453_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_1454_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N3: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N3 ) @ N3 ) ) ) ) ).

% modulo_nat_def
thf(fact_1455_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_1456_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_1457_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_1458_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_1459_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1460_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1461_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1462_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_1463_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1464_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1465_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1466_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_1467_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_1468_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M3: nat] :
              ( ( ( some_nat @ M3 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_1469_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_1470_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_1471_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_1472_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_1473_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1474_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1475_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1476_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1477_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1478_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z3: real] :
              ( ( power_power_real @ Z3 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_1479_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_1480_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y2: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_1481_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y2: nat] :
            ( X
            = ( some_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_1482_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y2: num] :
            ( X
            = ( some_num @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_1483_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y2: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y2 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_1484_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y2: nat] :
            ( X
           != ( some_nat @ Y2 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_1485_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y2: num] :
            ( X
           != ( some_num @ Y2 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_1486_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_1487_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_1488_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_1489_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_1490_mint__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_1491_maxt__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_1492_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_1493_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_1494_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_1495_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_1496_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_1497_finite__Collect__disjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        & ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1498_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X2: set_nat] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1499_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1500_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1501_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] :
              ( ( P @ X2 )
              | ( Q @ X2 ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1502_finite__Collect__conjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        | ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1503_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X2: set_nat] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1504_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1505_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1506_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] :
              ( ( P @ X2 )
              & ( Q @ X2 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1507_mint__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_1508_mint__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_1509_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_1510_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_1511_misiz,axiom,
    ! [T: vEBT_VEBT,N2: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% misiz
thf(fact_1512_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_1513_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_1514_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).

% lesseq_shift
thf(fact_1515_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_1516_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N: nat] :
        ! [X5: list_o] :
          ( ( member_list_o @ X5 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_1517_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N: nat] :
        ! [X5: list_int] :
          ( ( member_list_int @ X5 @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_1518_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_nat,R: real > nat > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1519_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_complex,R: real > complex > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B3 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1520_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_int,R: real > int > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1521_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1522_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_complex,R: nat > complex > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1523_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1524_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B3: set_nat,R: complex > nat > $o] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite3207457112153483333omplex
                  @ ( collect_complex
                    @ ^ [A3: complex] :
                        ( ( member_complex @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1525_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B3: set_complex,R: complex > complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B3 )
              & ~ ( finite3207457112153483333omplex
                  @ ( collect_complex
                    @ ^ [A3: complex] :
                        ( ( member_complex @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1526_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B3: set_int,R: complex > int > $o] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ~ ( finite3207457112153483333omplex
                  @ ( collect_complex
                    @ ^ [A3: complex] :
                        ( ( member_complex @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1527_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B3: set_nat,R: int > nat > $o] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X3 @ Xa ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A2 )
                        & ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1528_not__finite__existsD,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1529_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1530_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1531_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_1: complex] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1532_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_1533_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( ord_less_eq_real @ A @ X3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1534_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
            & ( ord_less_eq_set_nat @ A @ X3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1535_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1536_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1537_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1538_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1539_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( ord_less_eq_real @ X3 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1540_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
            & ( ord_less_eq_set_nat @ X3 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1541_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1542_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1543_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1544_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1545_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B6: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1546_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B6: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1547_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B6: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1548_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B6: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1549_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B6: nat] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1550_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B6: num] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1551_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B6: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1552_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B6: nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1553_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B6: num] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_num @ B6 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1554_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X2: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_1555_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X6: option_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X2: nat] : ( P4 @ ( some_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_1556_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X2: num] : ( P4 @ ( some_num @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_1557_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X2: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_1558_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X6: option_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X2: nat] : ( P4 @ ( some_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_1559_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X2: num] : ( P4 @ ( some_num @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_1560_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1561_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1562_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_1563_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_1564_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_1565_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_1566_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_1567_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_1568_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_1569_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_1570_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_1571_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_1572_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1573_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1574_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1575_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_1576_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_1577_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_1578_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_1579_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).

% less_shift
thf(fact_1580_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y2: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).

% greater_shift
thf(fact_1581_helpypredd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpypredd
thf(fact_1582_helpyd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpyd
thf(fact_1583_pred__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_1584_succ__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_1585_succ__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_1586_pred__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_1587_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N2 )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_1588_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_1589_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_1590_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_1591_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_1592_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_1593_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_1594_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_1595_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_1596_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_1597_pred__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ Y2 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_1598_succ__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ X @ Y2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_1599_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_1600_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_1601_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1602_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1603_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_1604_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_1605_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1606_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_1607_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_1608_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).

% buildup_gives_valid
thf(fact_1609_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1610_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1611_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1612_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1613_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( times_times_complex @ A @ C )
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1614_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1615_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1616_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_1617_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_1618_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( times_times_complex @ C @ A )
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1619_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1620_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1621_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_1622_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_1623_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_1624_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1625_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1626_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_1627_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_1628_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_1629_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1630_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1631_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1632_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1633_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_1634_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1635_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1636_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1637_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1638_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_1639_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1640_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1641_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1642_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1643_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_1644_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_1645_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_1646_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_1647_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_1648_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_1649_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_1650_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ B @ A ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_1651_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_1652_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_1653_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_1654_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_1655_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_1656_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_1657_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_1658_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_1659_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_1660_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_1661_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_1662_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_1663_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_1664_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_1665_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1666_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1667_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1668_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_1669_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1670_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1671_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1672_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1673_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_1674_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1675_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1676_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1677_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_1678_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1679_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1680_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1681_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1682_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_1683_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1684_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1685_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1686_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1687_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1688_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1689_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1690_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_1691_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_1692_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1693_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1694_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1695_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_1696_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_1697_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_1698_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_1699_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_1700_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_1701_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1702_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1703_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1704_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1705_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1706_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1707_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1708_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1709_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1710_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1711_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1712_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1713_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1714_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1715_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1716_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1717_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1718_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1719_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1720_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1721_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1722_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1723_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1724_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1725_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1726_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1727_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_1728_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_1729_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_1730_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1731_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_1732_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_1733_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1734_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1735_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1736_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1737_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1738_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1739_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1740_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1741_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1742_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1743_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_1744_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_1745_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_1746_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_1747_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_1748_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_1749_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_1750_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_1751_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_1752_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_1753_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_1754_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_1755_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_1756_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_1757_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_1758_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_1759_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1760_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1761_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_1762_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1763_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1764_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_1765_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1766_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1767_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_1768_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1769_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1770_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_1771_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_1772_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_1773_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_1774_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_1775_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_1776_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_1777_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_1778_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_1779_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_1780_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_1781_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_1782_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_1783_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_1784_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_1785_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_1786_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_1787_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1788_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1789_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_1790_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1791_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1792_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_1793_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_1794_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_1795_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1796_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_1797_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_1798_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_1799_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1800_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_1801_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_1802_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_1803_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1804_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_1805_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_1806_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_1807_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1808_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_1809_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1810_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1811_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_1812_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1813_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1814_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1815_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1816_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1817_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1818_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1819_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1820_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1821_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1822_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1823_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1824_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1825_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1826_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1827_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1828_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1829_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1830_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1831_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1832_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_1833_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1834_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_1835_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1836_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1837_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1838_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1839_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1840_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1841_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1842_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1843_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1844_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1845_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1846_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1847_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_1848_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_1849_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_1850_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_1851_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_1852_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_1853_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_1854_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_1855_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_1856_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_1857_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_1858_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_1859_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1860_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1861_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1862_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B ) )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1863_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1864_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1865_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = one_one_complex )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1866_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1867_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1868_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1869_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1870_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1871_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1872_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1873_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1874_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1875_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1876_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1877_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1878_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1879_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1880_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1881_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1882_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1883_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_1884_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_1885_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_1886_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_1887_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_1888_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_1889_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_1890_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_1891_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_1892_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1893_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1894_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_1895_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1896_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1897_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1898_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1899_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_1900_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_1901_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_1902_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_1903_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_1904_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_1905_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_1906_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1907_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_1908_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1909_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1910_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_1911_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1912_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_1913_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_1914_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1915_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1916_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1917_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1918_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1919_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1920_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1921_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1922_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = ( divide_divide_nat @ M @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_1923_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_1924_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1925_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1926_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_1927_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_1928_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1929_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1930_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1931_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1932_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1933_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1934_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1935_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1936_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1937_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1938_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_1939_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_1940_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1941_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1942_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1943_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1944_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1945_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1946_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1947_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1948_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1949_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1950_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1951_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1952_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_1953_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_1954_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_1955_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_1956_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_1957_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_1958_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_1959_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_1960_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( power_power_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1961_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1962_power__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( power_power_real @ A @ N2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1963_power__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( power_power_complex @ A @ N2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1964_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ( power_power_int @ A @ N2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1965_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_1966_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_1967_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_1968_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1969_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_1970_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_1971_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1972_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1973_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1974_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1975_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1976_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1977_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1978_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1979_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1980_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1981_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1982_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1983_power__mono__iff,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1984_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1985_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1986_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1987_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_1988_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_1989_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_1990_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_1991_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_1992_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_1993_bits__1__div__2,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% bits_1_div_2
thf(fact_1994_bits__1__div__2,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% bits_1_div_2
thf(fact_1995_one__div__two__eq__zero,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% one_div_two_eq_zero
thf(fact_1996_one__div__two__eq__zero,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% one_div_two_eq_zero
thf(fact_1997_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1998_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1999_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2000_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2001_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2002_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2003_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2004_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2005_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_2006_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_2007_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_2008_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_2009_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_2010_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_2011_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2012_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2013_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2014_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2015_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2016_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2017_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2018_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2019_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2020_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_2021_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_2022_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_2023_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_2024_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_2025_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_2026_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_2027_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_2028_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2029_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2030_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_2031_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_2032_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2033_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_rat @ zero_zero_rat @ N2 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_2034_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_nat @ zero_zero_nat @ N2 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_2035_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ zero_zero_real @ N2 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_2036_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ zero_zero_complex @ N2 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_2037_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_int @ zero_zero_int @ N2 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_2038_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_2039_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_2040_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_2041_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2042_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2043_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2044_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2045_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2046_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2047_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2048_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_2049_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_2050_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_2051_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_2052_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_2053_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2054_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2055_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_2056_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2057_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2058_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2059_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2060_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2061_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = ( times_times_complex @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2062_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2063_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2064_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_2065_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_2066_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A )
          = ( times_times_complex @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2067_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2068_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2069_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_2070_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_2071_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( times_times_complex @ A @ B )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_2072_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2073_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_2074_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_2075_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_2076_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_2077_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2078_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_2079_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_2080_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_2081_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_2082_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2083_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_2084_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_2085_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_2086_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2087_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2088_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2089_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2090_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2091_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2092_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2093_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2094_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2095_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_2096_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2097_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2098_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2099_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2100_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2101_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2102_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2103_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_2104_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_2105_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: complex,Z5: complex] : Y5 = Z5 )
    = ( ^ [A3: complex,B2: complex] :
          ( ( minus_minus_complex @ A3 @ B2 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2106_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z5: real] : Y5 = Z5 )
    = ( ^ [A3: real,B2: real] :
          ( ( minus_minus_real @ A3 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2107_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
    = ( ^ [A3: rat,B2: rat] :
          ( ( minus_minus_rat @ A3 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2108_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [A3: int,B2: int] :
          ( ( minus_minus_int @ A3 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2109_power__not__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N2 )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_2110_power__not__zero,axiom,
    ! [A: nat,N2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N2 )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_2111_power__not__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N2 )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_2112_power__not__zero,axiom,
    ! [A: complex,N2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N2 )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_2113_power__not__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N2 )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_2114_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_2115_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_2116_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_2117_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_2118_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_2119_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_2120_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_2121_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_2122_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_2123_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_2124_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_2125_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_2126_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_2127_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va3: nat] :
              ( X
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_2128_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ~ ( P @ N )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_2129_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_2130_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_2131_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_2132_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_2133_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_2134_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_2135_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_2136_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_2137_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_2138_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2139_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_2140_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_2141_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_2142_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_2143_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2144_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2145_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N2 )
              = ( power_power_real @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2146_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N2 )
              = ( power_power_rat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2147_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2148_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2149_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ N2 )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2150_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N2 )
        = ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2151_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2152_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2153_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_2154_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_2155_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2156_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2157_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2158_power__strict__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2159_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2160_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2161_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2162_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2163_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2164_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2165_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2166_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2167_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2168_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2169_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2170_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2171_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2172_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2173_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2174_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_2175_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_2176_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_2177_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_2178_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2179_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2180_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2181_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2182_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2183_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2184_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_2185_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_2186_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2187_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2188_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_2189_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2190_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2191_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_2192_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_2193_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2194_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2195_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_2196_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_2197_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2198_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2199_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_2200_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_2201_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2202_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2203_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_2204_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_2205_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2206_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2207_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_2208_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_2209_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2210_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2211_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_2212_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2213_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2214_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_2215_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_2216_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2217_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2218_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_2219_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2220_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2221_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_2222_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_2223_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2224_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2225_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_2226_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2227_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2228_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_2229_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2230_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2231_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_2232_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2233_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2234_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2235_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2236_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2237_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_2238_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_2239_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2240_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2241_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_2242_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_2243_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_2244_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_2245_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_2246_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_2247_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_2248_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_2249_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_2250_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_2251_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% zero_less_one_class.zero_le_one
thf(fact_2252_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one_class.zero_le_one
thf(fact_2253_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_2254_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_2255_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2256_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2257_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_2258_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_2259_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2260_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2261_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2262_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2263_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2264_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2265_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_2266_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_2267_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2268_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2269_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_2270_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_2271_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2272_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2273_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_2274_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_2275_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2276_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2277_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_2278_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_2279_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2280_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2281_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_2282_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_2283_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_2284_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2285_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_2286_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_2287_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2288_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2289_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_2290_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_2291_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2292_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2293_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_2294_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_2295_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2296_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2297_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_2298_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_2299_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2300_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2301_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_2302_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2303_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2304_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_2305_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_2306_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2307_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2308_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_2309_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2310_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2311_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_2312_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2313_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2314_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_2315_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_2316_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2317_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2318_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_2319_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2320_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2321_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_2322_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2323_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2324_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_2325_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2326_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2327_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_2328_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_2329_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2330_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2331_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_2332_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_2333_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2334_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2335_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_2336_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2337_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2338_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_2339_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_2340_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2341_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2342_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_2343_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_2344_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2345_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2346_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_2347_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_2348_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2349_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2350_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_2351_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_2352_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2353_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2354_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_2355_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2356_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2357_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2358_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2359_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2360_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_2361_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2362_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2363_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2364_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2365_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2366_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2367_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2368_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2369_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2370_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2371_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2372_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2373_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2374_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2375_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_2376_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2377_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2378_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_2379_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_2380_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_2381_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2382_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2383_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_2384_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_2385_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_2386_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_2387_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_2388_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_2389_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2390_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2391_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_2392_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2393_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2394_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2395_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2396_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2397_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2398_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2399_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2400_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2401_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2402_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2403_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2404_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2405_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2406_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2407_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2408_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2409_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2410_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_2411_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2412_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2413_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2414_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2415_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2416_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2417_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2418_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2419_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2420_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2421_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2422_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2423_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2424_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2425_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2426_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2427_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2428_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2429_power__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2430_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2431_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2432_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2433_zero__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2434_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2435_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2436_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2437_zero__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2438_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2439_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2440_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2441_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2442_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2443_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2444_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B @ C )
          = A )
        = ( B
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2445_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2446_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2447_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B )
       => ( A
          = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2448_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2449_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2450_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2451_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2452_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2453_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_2454_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2455_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2456_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_2457_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2458_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2459_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X @ Z )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2460_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2461_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2462_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = one_one_complex )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2463_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2464_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2465_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2466_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2467_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2468_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2469_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2470_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2471_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2472_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2473_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2474_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2475_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2476_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2477_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2478_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_2479_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_2480_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_2481_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_2482_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M5: nat] :
            ( N2
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2483_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_2484_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2485_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2486_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2487_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2488_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2489_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2490_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_2491_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2492_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2493_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2494_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2495_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_2496_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_2497_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_2498_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2499_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2500_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2501_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_2502_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_2503_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2504_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
          = N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
         != N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_2505_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_2506_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q2: nat] :
          ( M
          = ( times_times_nat @ D @ Q2 ) ) ) ).

% mod_eq_0D
thf(fact_2507_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_2508_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_2509_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_2510_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A5: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A5 ) )
               => ! [B6: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B6 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A5 @ B6 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_2511_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: num > num > num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A5: num] :
                ( ( Xa2
                  = ( some_num @ A5 ) )
               => ! [B6: num] :
                    ( ( Xb
                      = ( some_num @ B6 ) )
                   => ( Y
                     != ( some_num @ ( X @ A5 @ B6 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_2512_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A5: nat] :
                ( ( Xa2
                  = ( some_nat @ A5 ) )
               => ! [B6: nat] :
                    ( ( Xb
                      = ( some_nat @ B6 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A5 @ B6 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_2513_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_2514_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_2515_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2516_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2517_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_2518_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2519_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2520_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_2521_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2522_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2523_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_2524_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_2525_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2526_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2527_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_2528_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_2529_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2530_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2531_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_2532_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2533_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2534_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_2535_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_2536_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2537_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2538_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_2539_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_2540_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2541_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2542_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_2543_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2544_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2545_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_2546_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2547_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2548_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_2549_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2550_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2551_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_2552_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_2553_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2554_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2555_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_2556_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_2557_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2558_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2559_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_2560_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_2561_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2562_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2563_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_2564_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_2565_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2566_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2567_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2568_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2569_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_2570_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_2571_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2572_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2573_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_2574_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_2575_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_2576_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_2577_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_2578_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_2579_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_2580_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_2581_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_2582_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_2583_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_2584_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_2585_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_2586_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_2587_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_2588_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_2589_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_2590_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_2591_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2592_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2593_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_2594_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2595_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2596_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_2597_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2598_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_2599_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_2600_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_2601_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2602_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2603_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_2604_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_2605_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2606_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2607_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2608_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2609_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2610_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2611_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2612_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2613_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2614_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2615_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2616_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2617_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2618_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2619_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2620_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2621_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2622_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2623_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2624_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2625_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2626_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2627_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_2628_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2629_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2630_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_2631_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2632_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2633_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_2634_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2635_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2636_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_2637_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2638_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2639_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2640_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2641_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2642_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2643_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2644_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2645_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2646_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2647_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2648_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2649_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2650_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2651_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2652_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2653_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2654_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2655_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2656_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2657_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2658_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2659_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2660_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2661_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2662_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2663_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2664_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2665_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2666_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2667_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2668_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2669_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2670_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2671_power__le__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2672_power__le__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2673_power__le__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2674_power__le__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2675_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2676_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2677_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2678_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2679_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2680_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2681_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2682_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2683_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2684_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2685_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2686_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2687_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2688_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2689_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2690_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2691_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2692_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2693_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2694_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2695_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2696_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2697_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2698_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2699_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2700_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2701_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2702_power__inject__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N2 ) )
        = ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2703_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
        = ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2704_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2705_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2706_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2707_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2708_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2709_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2710_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2711_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2712_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2713_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2714_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2715_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2716_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2717_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2718_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2719_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2720_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2721_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2722_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2723_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2724_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2725_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2726_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2727_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2728_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2729_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2730_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2731_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2732_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2733_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2734_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2735_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_2736_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_2737_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_2738_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_2739_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_2740_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_2741_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2742_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_2743_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_2744_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_2745_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2746_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2747_length__pos__if__in__set,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2748_length__pos__if__in__set,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2749_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2750_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2751_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2752_length__pos__if__in__set,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2753_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2754_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2755_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_2756_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2757_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_2758_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_2759_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_2760_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_2761_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N2 )
          = M )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2762_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_2763_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2764_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2765_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_2766_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_2767_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_2768_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_2769_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_2770_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_2771_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( ( ord_less_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2772_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2773_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2774_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2775_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_2776_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2777_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2778_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_2779_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2780_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2781_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_2782_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2783_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2784_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_2785_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2786_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2787_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_2788_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2789_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2790_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_2791_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2792_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2793_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_2794_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2795_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2796_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_2797_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2798_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2799_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2800_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2801_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2802_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2803_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2804_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2805_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2806_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2807_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2808_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2809_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2810_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2811_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2812_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2813_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2814_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2815_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2816_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2817_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2818_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2819_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2820_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2821_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2822_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2823_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_2824_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2825_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2826_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2827_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2828_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2829_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2830_power__Suc__less,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2831_power__Suc__less,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2832_power__Suc__less,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2833_power__Suc__less,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2834_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2835_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2836_power__Suc__le__self,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2837_power__Suc__le__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2838_power__Suc__le__self,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2839_power__Suc__le__self,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2840_power__Suc__less__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2841_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2842_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2843_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2844_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2845_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2846_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2847_power__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N5 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2848_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2849_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2850_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2851_power__decreasing,axiom,
    ! [N2: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N5 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2852_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2853_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2854_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2855_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2856_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2857_self__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2858_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2859_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2860_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2861_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_2862_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_2863_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_2864_one__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2865_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2866_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2867_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2868_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_2869_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2870_power__diff,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2871_power__diff,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2872_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2873_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2874_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_2875_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_2876_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_2877_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_2878_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N3 )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).

% div_if
thf(fact_2879_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_2880_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_2881_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2882_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2883_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_2884_split__div,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_2885_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_2886_split__mod,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_2887_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_2888_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_2889_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_2890_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_2891_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_2892_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_2893_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2894_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2895_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_2896_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_2897_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_2898_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_2899_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_2900_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_2901_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_2902_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2903_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2904_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2905_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2906_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2907_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2908_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2909_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2910_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2911_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2912_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2913_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2914_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2915_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2916_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2917_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_2918_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_2919_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_2920_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2921_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2922_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2923_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2924_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2925_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2926_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2927_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2928_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2929_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_2930_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_2931_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_2932_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_2933_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_2934_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_2935_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2936_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2937_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_2938_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_2939_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_2940_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2941_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2942_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2943_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2944_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2945_power__minus__mult,axiom,
    ! [N2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2946_power__minus__mult,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2947_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2948_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2949_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2950_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_2951_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_2952_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_2953_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_2954_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_2955_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2956_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2957_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2958_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2959_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2960_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2961_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2962_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2963_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2964_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2965_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2966_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2967_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2968_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2969_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_2970_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2971_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2972_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2973_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2974_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_2975_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_2976_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_2977_zero__le__even__power_H,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2978_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2979_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2980_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_2981_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_2982_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_2983_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A5: product_prod_nat_nat,B6: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A5 ) @ ( some_P7363390416028606310at_nat @ B6 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_2984_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A5: nat,B6: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A5 ) @ ( some_nat @ B6 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_2985_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A5: num,B6: num] :
              ( X
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A5 ) @ ( some_num @ B6 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_2986_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_2987_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X3: nat,Y3: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_2988_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X3: num,Y3: num] :
              ( X
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_2989_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_2990_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2991_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2992_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2993_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2994_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2995_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2996_odd__power__less__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_2997_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_2998_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_2999_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_3000_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_3001_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_3002_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3003_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3004_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3005_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3006_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3007_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_3008_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3009_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3010_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3011_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3012_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3013_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_3014_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_3015_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_3016_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_3017_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X )
                    & ( ord_less_nat @ X @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_3018_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu2: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B6 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B6
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B6
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N: nat] :
                  ( Xa2
                  = ( suc @ N ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_3019_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A5: $o,B6: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B6 ) )
               => ( ? [Va3: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B6
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B6
                       => ( ( A5
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A5
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_3020_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_3021_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_3022_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3023_inrange,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3024_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa2 )
        = Y )
     => ( ( ? [A5: $o,B6: $o] :
              ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
         => ( Y
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                        @ ( if_nat
                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                            & ~ ( ( Xa2 = Mi2 )
                                | ( Xa2 = Ma2 ) ) )
                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                          @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_3025_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_3026_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3027_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3028_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_3029_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A3: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).

% deg1Leaf
thf(fact_3030_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A5: $o,B6: $o] :
          ( T
          = ( vEBT_Leaf @ A5 @ B6 ) ) ) ).

% deg_1_Leaf
thf(fact_3031_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A5: $o,B6: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B6 ) ) ) ) ).

% deg_1_Leafy
thf(fact_3032_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_3033_verit__eq__simplify_I9_J,axiom,
    ! [X33: num,Y32: num] :
      ( ( ( bit1 @ X33 )
        = ( bit1 @ Y32 ) )
      = ( X33 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_3034_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_3035_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% VEBT.inject(2)
thf(fact_3036_i0__less,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_3037_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_3038_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = N2 ) ).

% idiff_0_right
thf(fact_3039_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_3040_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_3041_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_3042_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_3043_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_3044_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_3045_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_3046_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_3047_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_3048_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_3049_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_3050_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_3051_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_3052_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_3053_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_3054_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_3055_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_3056_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_3057_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_3058_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_3059_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_3060_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).

% enat_0_less_mult_iff
thf(fact_3061_not__iless0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_3062_iadd__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_3063_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_3064_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).

% i0_lb
thf(fact_3065_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,N2: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A2 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B3 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_3066_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).

% set_bit_greater_eq
thf(fact_3067_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N2 )
           => ( P @ M5 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_3068_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N2 )
            & ( P @ M5 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_3069_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3070_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_3071_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu @ $true ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_3072_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_3073_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
    = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
thf(fact_3074_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_3075_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_3076_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3077_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_3078_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_3079_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_3080_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_3081_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_3082_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_3083_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3084_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_3085_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_3086_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_3087_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_3088_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_3089_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_3090_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_3091_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_3092_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_3093_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_3094_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_3095_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q2: int] :
          ( M
          = ( times_times_int @ D @ Q2 ) ) ) ).

% zmod_eq_0D
thf(fact_3096_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M
            = ( times_times_int @ D @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_3097_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B6: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ X3 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3098_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_3099_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_3100_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_3101_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3102_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_3103_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_3104_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_3105_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_3106_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3107_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3108_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3109_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3110_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3111_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_3112_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3113_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3114_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3115_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3116_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3117_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_3118_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3119_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_3120_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_3121_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3122_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N: nat] : ( ord_less_real @ ( power_power_real @ X @ N ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3123_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3124_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3125_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B4: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3126_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B4: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3127_q__pos__lemma,axiom,
    ! [B4: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3128_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_3129_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_3130_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A5: $o,B6: $o] :
          ( X
         != ( vEBT_Leaf @ A5 @ B6 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_3131_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_3132_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,N2: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_3133_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3134_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_3135_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N2 )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N2 )
                    = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3136_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_3137_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_3138_split__zdiv,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( divide_divide_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_3139_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3140_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3141_split__zmod,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_3142_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3143_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3144_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_3145_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_3146_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_3147_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_3148_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_3149_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_3150_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_3151_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_3152_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_3153_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_3154_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,Uw2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B6: $o,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_3155_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B6: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B6 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_3156_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X3 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3157_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B6: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X3 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_3158_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B6: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_3159_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_3160_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ( ~ ( ord_less_eq_real @ B4 @ A4 ) )
      = ( ord_less_real @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3161_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ( ~ ( ord_less_eq_rat @ B4 @ A4 ) )
      = ( ord_less_rat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3162_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ( ~ ( ord_less_eq_num @ B4 @ A4 ) )
      = ( ord_less_num @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3163_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
      = ( ord_less_nat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3164_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
      = ( ord_less_int @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3165_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_3166_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_3167_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_3168_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_3169_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_3170_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_3171_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(14)
thf(fact_3172_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] :
      ( one
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(12)
thf(fact_3173_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_3174_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_3175_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X
        = ( minus_minus_real @ Y @ Z ) )
      = ( Y
        = ( plus_plus_real @ X @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_3176_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_3177_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3178_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3179_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_3180_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_3181_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ~ ( ( A5
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A5
                 => ( ( B6
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B6
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_3182_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ~ ( ( B6
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B6
                 => ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_3183_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_3184_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa2 )
        = Y )
     => ( ( ? [A5: $o,B6: $o] :
              ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ one_one_nat
                        @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa2 )
                                    & ( ord_less_nat @ Xa2 @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                  @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_3185_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B6 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_3186_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B6 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_3187_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B6 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3188_div__less__mono,axiom,
    ! [A2: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B3 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_3189_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3190_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3191_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3192_div__mod__decomp,axiom,
    ! [A2: nat,N2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_3193_div__mod__decomp__int,axiom,
    ! [A2: int,N2: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_3194_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B6 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_3195_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3196_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3197_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B6 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_3198_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B6 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_3199_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa2 )
        = Y )
     => ( ( ? [A5: $o,B6: $o] :
              ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
         => ( Y
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_3200_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa2 )
        = Y )
     => ( ( ? [A5: $o,B6: $o] :
              ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( if_nat
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                        @ one_one_nat ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_3201_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A5: $o,B6: $o] :
              ( A1
              = ( vEBT_Leaf @ A5 @ B6 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X5 @ N ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                     => ( ( M3 = N )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N @ M3 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X5 @ N ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                       => ( ( M3
                            = ( suc @ N ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N @ M3 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X5 @ N ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                         => ( ( M3 = N )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N @ M3 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X4 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                & ! [X5: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X5 )
                                                      & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X5 @ N ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                           => ( ( M3
                                = ( suc @ N ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N @ M3 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X4 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                  & ! [X5: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X5 )
                                                        & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3202_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A3: $o,B2: $o] :
                ( A12
                = ( vEBT_Leaf @ A3 @ B2 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                 => ( vEBT_invar_vebt @ X2 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
          | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList2 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                 => ( vEBT_invar_vebt @ X2 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
          | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                 => ( vEBT_invar_vebt @ X2 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X4 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList2: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList2 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                 => ( vEBT_invar_vebt @ X2 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X4 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3203_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_3204_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_3205_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_3206_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_3207_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_3208_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_3209_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_3210_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3211_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3212_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3213_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3214_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3215_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3216_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3217_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3218_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3219_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3220_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3221_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3222_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3223_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3224_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3225_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3226_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3227_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3228_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ! [A5: $o] :
            ( ? [B6: $o] :
                ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A5 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_3229_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B6 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B6
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B6
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B6 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N: nat] :
                    ( ( Xa2
                      = ( suc @ N ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_3230_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B6: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B6 ) )
                 => ! [Va3: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B6
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B6
                           => ( ( A5
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A5
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_3231_Icc__eq__Icc,axiom,
    ! [L: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H2 )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H2 )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3232_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3233_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3234_Icc__eq__Icc,axiom,
    ! [L: nat,H2: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H2 )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3235_Icc__eq__Icc,axiom,
    ! [L: int,H2: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H2 )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H2 )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3236_Icc__eq__Icc,axiom,
    ! [L: real,H2: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H2 )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L @ H2 )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3237_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3238_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3239_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3240_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3241_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3242_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3243_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_3244_imult__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_3245_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_3246_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M7: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M7 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3247_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc3368934014287244435at_num] :
      ~ ! [F2: nat > num > num,A5: nat,B6: nat,Acc: num] :
          ( X
         != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A5 @ ( product_Pair_nat_num @ B6 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_3248_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A5: nat,B6: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B6 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_3249_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_3250_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N6 )
         => ( ord_less_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_3251_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_nat @ X3 @ N2 ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_3252_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N6 )
         => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_3253_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3254_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3255_mint__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% mint_bound
thf(fact_3256_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_3257_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_3258_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3259_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3260_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3261_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3262_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3263_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3264_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3265_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3266_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                & ~ ( ( Xa2 = Mi2 )
                                    | ( Xa2 = Ma2 ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                              @ one_one_nat ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_3267_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( if_nat
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                            @ one_one_nat ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_3268_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa2 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa2 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_3269_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( Xa2 = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa2 = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa2 ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa2 )
                                        & ( ord_less_nat @ Xa2 @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                      @ zero_zero_nat ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_3270_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B6 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3271_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B6 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3272_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B6 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3273_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B6 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_3274_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B6 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_3275_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B6 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_3276_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_3277_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_3278_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_3279_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B3 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X4: int] : ( P @ X4 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ B3 )
                        & ( P @ ( plus_plus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3280_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A2 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X4: int] : ( P @ X4 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ A2 )
                        & ( P @ ( minus_minus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3281_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_3282_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_3283_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ( ? [X4: int] : ( P @ X4 ) )
          = ( ? [X2: int] :
                ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3284_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_3285_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_3286_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_3287_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_3288_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_3289_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3290_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3291_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3292_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3293_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3294_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3295_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3296_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3297_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3298_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3299_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3300_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3301_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3302_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3303_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3304_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3305_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3306_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3307_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3308_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3309_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_real @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3310_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_rat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3311_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_num @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3312_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_nat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3313_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_int @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3314_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_real @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3315_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_rat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3316_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_num @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3317_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_nat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3318_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_int @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3319_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3320_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3321_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3322_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3323_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3324_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3325_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3326_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3327_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3328_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3329_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3330_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3331_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3332_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3333_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3334_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3335_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3336_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3337_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3338_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3339_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_real @ X5 @ T ) ) ).

% minf(5)
thf(fact_3340_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_rat @ X5 @ T ) ) ).

% minf(5)
thf(fact_3341_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_num @ X5 @ T ) ) ).

% minf(5)
thf(fact_3342_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_nat @ X5 @ T ) ) ).

% minf(5)
thf(fact_3343_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_int @ X5 @ T ) ) ).

% minf(5)
thf(fact_3344_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_real @ T @ X5 ) ) ).

% minf(7)
thf(fact_3345_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_rat @ T @ X5 ) ) ).

% minf(7)
thf(fact_3346_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_num @ T @ X5 ) ) ).

% minf(7)
thf(fact_3347_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_nat @ T @ X5 ) ) ).

% minf(7)
thf(fact_3348_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_int @ T @ X5 ) ) ).

% minf(7)
thf(fact_3349_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3350_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3351_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_eq_num @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3352_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3353_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_eq_int @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3354_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3355_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_eq_rat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3356_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_eq_num @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3357_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3358_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_eq_int @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3359_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_3360_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_eq_rat @ X5 @ T ) ) ).

% minf(6)
thf(fact_3361_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_eq_num @ X5 @ T ) ) ).

% minf(6)
thf(fact_3362_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_3363_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_eq_int @ X5 @ T ) ) ).

% minf(6)
thf(fact_3364_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_3365_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).

% minf(8)
thf(fact_3366_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_eq_num @ T @ X5 ) ) ).

% minf(8)
thf(fact_3367_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_3368_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_eq_int @ T @ X5 ) ) ).

% minf(8)
thf(fact_3369_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K3: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X3: real,K3: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
       => ! [X5: real,K4: real] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3370_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K3: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X3: rat,K3: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X5: rat,K4: rat] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3371_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K3: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X3: int,K3: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
       => ! [X5: int,K4: int] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3372_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K3: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X3: real,K3: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
       => ! [X5: real,K4: real] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3373_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K3: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X3: rat,K3: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X5: rat,K4: rat] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3374_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K3: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X3: int,K3: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
       => ! [X5: int,K4: int] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3375_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3376_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3377_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_3378_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_3379_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_3380_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_3381_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_3382_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3383_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z4 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_3384_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z4 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_3385_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3386_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3387_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B6 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_3388_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3389_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3390_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3391_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3392_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3393_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3394_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3395_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3396_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A5: real,B6: real,C2: real] :
            ( ( P @ A5 @ B6 )
           => ( ( P @ B6 @ C2 )
             => ( ( ord_less_eq_real @ A5 @ B6 )
               => ( ( ord_less_eq_real @ B6 @ C2 )
                 => ( P @ A5 @ C2 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A5: real,B6: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B6 )
                          & ( ord_less_real @ ( minus_minus_real @ B6 @ A5 ) @ D5 ) )
                       => ( P @ A5 @ B6 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_3397_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_3398_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_3399_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_3400_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_3401_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_3402_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_3403_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_3404_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_3405_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_3406_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_3407_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_3408_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_3409_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_3410_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3411_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3412_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3413_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_3414_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_3415_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_3416_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_3417_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_int_def
thf(fact_3418_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_3419_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_3420_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_3421_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_3422_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_3423_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_3424_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% maxt_bound
thf(fact_3425_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_3426_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M5 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M5 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_3427_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M5 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M5 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_3428_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_3429_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3430_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3431_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3432_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3433_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3434_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3435_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_3436_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_3437_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_3438_product__nth,axiom,
    ! [N2: nat,Xs2: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3439_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys: list_num] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3440_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3441_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) @ N2 )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3442_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_nat_o @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3443_product__nth,axiom,
    ! [N2: nat,Xs2: list_Code_integer,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs2 @ Ys ) @ N2 )
        = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3444_product__nth,axiom,
    ! [N2: nat,Xs2: list_nat,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_nat_int @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3445_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N2 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3446_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N2 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3447_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N2 )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3448_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3449_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3450_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3451_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3452_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3453_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3454_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3455_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3456_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3457_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3458_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3459_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3460_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3461_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3462_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3463_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3464_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3465_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3466_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3467_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3468_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3469_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3470_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3471_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3472_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_3473_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_3474_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_3475_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_3476_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_3477_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_3478_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_3479_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_3480_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3481_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3482_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3483_length__product,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3484_length__product,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3485_length__product,axiom,
    ! [Xs2: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3486_length__product,axiom,
    ! [Xs2: list_int,Ys: list_VEBT_VEBT] :
      ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3487_length__product,axiom,
    ! [Xs2: list_int,Ys: list_o] :
      ( ( size_s4246224855604898693_int_o @ ( product_int_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3488_length__product,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3489_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( Q3 = Q5 ) ) ) ).

% unique_quotient
thf(fact_3490_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_3491_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_3492_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q3 ) ) ).

% div_int_unique
thf(fact_3493_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_3494_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_3495_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_3496_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_3497_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_3498_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_3499_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_3500_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_3501_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3502_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3503_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3504_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_3505_prod_Oinject,axiom,
    ! [X1: code_integer,X22: $o,Y1: code_integer,Y22: $o] :
      ( ( ( produc6677183202524767010eger_o @ X1 @ X22 )
        = ( produc6677183202524767010eger_o @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3506_prod_Oinject,axiom,
    ! [X1: num,X22: num,Y1: num,Y22: num] :
      ( ( ( product_Pair_num_num @ X1 @ X22 )
        = ( product_Pair_num_num @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3507_prod_Oinject,axiom,
    ! [X1: nat,X22: num,Y1: nat,Y22: num] :
      ( ( ( product_Pair_nat_num @ X1 @ X22 )
        = ( product_Pair_nat_num @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3508_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3509_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( ( product_Pair_int_int @ X1 @ X22 )
        = ( product_Pair_int_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3510_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: $o,A4: code_integer,B4: $o] :
      ( ( ( produc6677183202524767010eger_o @ A @ B )
        = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3511_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A4: num,B4: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3512_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A4: nat,B4: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3513_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3514_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A4: int,B4: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3515_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6271795597528267376eger_o] :
      ~ ! [A5: code_integer,B6: $o] :
          ( Y
         != ( produc6677183202524767010eger_o @ A5 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_3516_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A5: num,B6: num] :
          ( Y
         != ( product_Pair_num_num @ A5 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_3517_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_num] :
      ~ ! [A5: nat,B6: num] :
          ( Y
         != ( product_Pair_nat_num @ A5 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_3518_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B6: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_3519_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A5: int,B6: int] :
          ( Y
         != ( product_Pair_int_int @ A5 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_3520_surj__pair,axiom,
    ! [P2: produc6271795597528267376eger_o] :
    ? [X3: code_integer,Y3: $o] :
      ( P2
      = ( produc6677183202524767010eger_o @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_3521_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X3: num,Y3: num] :
      ( P2
      = ( product_Pair_num_num @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_3522_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X3: nat,Y3: num] :
      ( P2
      = ( product_Pair_nat_num @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_3523_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_3524_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P2
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_3525_prod__cases,axiom,
    ! [P: produc6271795597528267376eger_o > $o,P2: produc6271795597528267376eger_o] :
      ( ! [A5: code_integer,B6: $o] : ( P @ ( produc6677183202524767010eger_o @ A5 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_3526_prod__cases,axiom,
    ! [P: product_prod_num_num > $o,P2: product_prod_num_num] :
      ( ! [A5: num,B6: num] : ( P @ ( product_Pair_num_num @ A5 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_3527_prod__cases,axiom,
    ! [P: product_prod_nat_num > $o,P2: product_prod_nat_num] :
      ( ! [A5: nat,B6: num] : ( P @ ( product_Pair_nat_num @ A5 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_3528_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A5: nat,B6: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_3529_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
      ( ! [A5: int,B6: int] : ( P @ ( product_Pair_int_int @ A5 @ B6 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_3530_Pair__inject,axiom,
    ! [A: code_integer,B: $o,A4: code_integer,B4: $o] :
      ( ( ( produc6677183202524767010eger_o @ A @ B )
        = ( produc6677183202524767010eger_o @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B = ~ B4 ) ) ) ).

% Pair_inject
thf(fact_3531_Pair__inject,axiom,
    ! [A: num,B: num,A4: num,B4: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3532_Pair__inject,axiom,
    ! [A: nat,B: num,A4: nat,B4: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3533_Pair__inject,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3534_Pair__inject,axiom,
    ! [A: int,B: int,A4: int,B4: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3535_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
      = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
thf(fact_3536_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_3537_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_3538_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_3539_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
     => ( ! [M3: nat,N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ( P @ N @ ( modulo_modulo_nat @ M3 @ N ) )
             => ( P @ M3 @ N ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_3540_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_3541_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3542_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3543_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3544_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3545_list__update__overwrite,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_3546_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_3547_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_3548_length__list__update,axiom,
    ! [Xs2: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I @ X ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_3549_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_3550_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_3551_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_3552_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_3553_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_3554_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_3555_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_3556_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_int,X: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
        = ( nth_int @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3557_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3558_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3559_list__update__id,axiom,
    ! [Xs2: list_int,I: nat] :
      ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3560_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3561_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3562_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_3563_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_3564_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_3565_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_3566_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_3567_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ V ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3568_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3569_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3570_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3571_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3572_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(4)
thf(fact_3573_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_3574_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_3575_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_3576_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_3577_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(3)
thf(fact_3578_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_3579_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_3580_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_3581_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_3582_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_3583_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_3584_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_3585_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_3586_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_3587_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_3588_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_3589_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_3590_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_3591_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_3592_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(5)
thf(fact_3593_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_3594_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_3595_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_3596_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_3597_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(6)
thf(fact_3598_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_3599_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_3600_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_3601_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_3602_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3603_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3604_list__update__beyond,axiom,
    ! [Xs2: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( list_update_int @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3605_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_3606_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_3607_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3608_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3609_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3610_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3611_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_3612_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_3613_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_3614_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_3615_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3616_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3617_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3618_set__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3619_list__update__swap,axiom,
    ! [I: nat,I6: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT,X7: vEBT_VEBT] :
      ( ( I != I6 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I6 @ X7 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I6 @ X7 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_3620_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3621_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3622_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3623_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3624_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3625_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3626_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3627_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3628_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3629_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3630_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3631_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_3632_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N2 @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_3633_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_3634_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_3635_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_3636_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3637_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A3: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3638_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3639_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3640_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3641_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3642_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_3643_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
      = ( ord_max_nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_3644_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A2: set_complex,X: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3645_set__update__subsetI,axiom,
    ! [Xs2: list_real,A2: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3646_set__update__subsetI,axiom,
    ! [Xs2: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3647_set__update__subsetI,axiom,
    ! [Xs2: list_int,A2: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3648_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3649_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_3650_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_3651_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_3652_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_3653_dbl__def,axiom,
    ( neg_nu7009210354673126013omplex
    = ( ^ [X2: complex] : ( plus_plus_complex @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_3654_semiring__norm_I27_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit0 @ N2 ) )
      = ( bit1 @ ( bitM @ N2 ) ) ) ).

% semiring_norm(27)
thf(fact_3655_semiring__norm_I28_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ N2 ) ) ) ).

% semiring_norm(28)
thf(fact_3656_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_complex,X: complex] :
      ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3657_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3658_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3659_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3660_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3661_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3662_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3663_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_3664_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_3665_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_3666_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_3667_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3668_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3669_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J )
        = ( ( ( I = J )
           => X )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3670_nth__list__update,axiom,
    ! [I: nat,Xs2: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3671_eval__nat__numeral_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_3672_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_3673_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_3674_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_3675_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bitM @ N2 ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N2 ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_3676_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N2 ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_3677_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bitM @ N2 ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_3678_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B6: nat] :
          ( ( P @ A5 @ B6 )
          = ( P @ B6 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B6: nat] :
              ( ( P @ A5 @ B6 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_3679_VEBT__internal_Oheight_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A5: $o,B6: $o] :
          ( X
         != ( vEBT_Leaf @ A5 @ B6 ) )
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_3680_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3681_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B6: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B6 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ B6 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_3682_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B6 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ B6 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B6 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_3683_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3684_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3685_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3686_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3687_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3688_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3689_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3690_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3691_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3692_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3693_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3694_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3695_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X @ Z )
        & ( ord_le72135733267957522d_enat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3696_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3697_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3698_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3699_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3700_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3701_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
      = ( ( ord_le2932123472753598470d_enat @ B @ A )
        & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3702_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3703_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3704_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3705_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3706_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3707_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3708_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3709_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3710_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3711_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3712_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3713_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3714_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3715_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3716_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = Q3 ) ).

% max_enat_simps(3)
thf(fact_3717_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = Q3 ) ).

% max_enat_simps(2)
thf(fact_3718_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K3: nat,M3: nat] :
          ( X
         != ( product_Pair_nat_nat @ K3 @ M3 ) ) ).

% prod_decode_aux.cases
thf(fact_3719_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] :
            ( X
           != ( suc @ N ) ) ) ).

% list_decode.cases
thf(fact_3720_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ( ord_le2932123472753598470d_enat @ D @ B )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3721_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3722_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3723_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3724_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3725_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3726_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3727_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_3728_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3729_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_3730_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_3731_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_3732_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_3733_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_3734_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_3735_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3736_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3737_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3738_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3739_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3740_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ A )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3741_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3742_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3743_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3744_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3745_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( A3
          = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3746_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A3: rat] :
          ( A3
          = ( ord_max_rat @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3747_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( A3
          = ( ord_max_num @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3748_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( A3
          = ( ord_max_nat @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3749_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( A3
          = ( ord_max_int @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3750_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3751_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3752_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_3753_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3754_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_3755_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3756_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3757_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_3758_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3759_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_3760_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X )
        | ( ord_le2932123472753598470d_enat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3761_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3762_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3763_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3764_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3765_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3766_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_max_rat @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3767_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_max_num @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3768_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_max_nat @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3769_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_max_int @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3770_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3771_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_max_rat @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3772_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_max_num @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3773_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_max_nat @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3774_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_max_int @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3775_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3776_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3777_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3778_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3779_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3780_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3781_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3782_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3783_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3784_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3785_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3786_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3787_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3788_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3789_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3790_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3791_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3792_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3793_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3794_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3795_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3796_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3797_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( A3
            = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3798_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( A3
            = ( ord_max_real @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3799_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( A3
            = ( ord_max_rat @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3800_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( A3
            = ( ord_max_num @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3801_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( A3
            = ( ord_max_nat @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3802_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( A3
            = ( ord_max_int @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3803_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3804_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3805_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3806_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3807_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3808_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3809_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X )
        | ( ord_le72135733267957522d_enat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3810_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3811_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3812_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3813_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3814_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3815_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_3816_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3817_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3818_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3819_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_3820_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_3821_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_3822_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3823_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3824_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3825_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3826_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X22: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3827_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X22: num] :
      ( ( size_option_num @ X @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3828_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_3829_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_3830_set__decode__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_3831_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_3832_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_3833_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_3834_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_3835_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_3836_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_3837_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_3838_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_3839_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_3840_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_3841_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_3842_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_3843_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_3844_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_3845_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_3846_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3847_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3848_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3849_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3850_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3851_dvd__add__triv__right__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ A ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3852_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3853_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3854_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3855_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3856_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3857_dvd__add__triv__left__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ A @ B ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3858_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3859_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3860_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3861_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_3862_signed__take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_3863_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3864_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3865_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3866_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3867_case__prod__conv,axiom,
    ! [F: int > int > int,A: int,B: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_3868_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3869_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3870_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3871_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3872_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3873_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3874_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3875_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3876_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3877_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3878_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3879_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3880_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3881_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3882_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3883_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3884_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_3885_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_3886_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_3887_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3888_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3889_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3890_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3891_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3892_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3893_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3894_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3895_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3896_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3897_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3898_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3899_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3900_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3901_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3902_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3903_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3904_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3905_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3906_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3907_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3908_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_3909_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_3910_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_3911_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_3912_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_3913_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_3914_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3915_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3916_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3917_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_3918_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_3919_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_3920_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_3921_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_3922_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_3923_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_3924_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_3925_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_Suc_Suc_iff
thf(fact_3926_even__Suc,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% even_Suc
thf(fact_3927_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3928_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3929_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3930_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3931_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3932_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3933_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3934_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3935_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3936_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3937_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3938_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3939_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3940_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3941_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3942_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3943_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3944_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_3945_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_3946_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_3947_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_3948_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_3949_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_3950_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_3951_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_3952_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_3953_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_3954_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3955_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3956_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3957_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3958_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3959_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3960_power__less__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_3961_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_3962_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_3963_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3964_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3965_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3966_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_3967_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_3968_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% odd_Suc_minus_one
thf(fact_3969_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_3970_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3971_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3972_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3973_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3974_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3975_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3976_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3977_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3978_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3979_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_3980_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_3981_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_3982_even__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3983_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3984_even__power,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3985_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_3986_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_3987_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_3988_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_3989_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3990_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3991_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3992_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3993_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3994_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3995_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3996_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3997_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3998_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_3999_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4000_prod_Ocase__distrib,axiom,
    ! [H2: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4001_prod_Ocase__distrib,axiom,
    ! [H2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4002_prod_Ocase__distrib,axiom,
    ! [H2: int > int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4003_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4004_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4005_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4006_prod_Ocase__distrib,axiom,
    ! [H2: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4007_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4008_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_4009_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4010_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4011_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4012_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_4013_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_4014_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_4015_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_4016_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_4017_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_4018_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_4019_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_4020_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_4021_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B2: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B2 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_4022_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B2: real] :
          ( ( A3 = zero_zero_real )
         => ( B2 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_4023_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( A3 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_4024_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_4025_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P2
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_4026_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B7: nat,C4: nat] :
          ( ( A
            = ( times_times_nat @ B7 @ C4 ) )
          & ( dvd_dvd_nat @ B7 @ B )
          & ( dvd_dvd_nat @ C4 @ C ) ) ) ).

% division_decomp
thf(fact_4027_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B7: int,C4: int] :
          ( ( A
            = ( times_times_int @ B7 @ C4 ) )
          & ( dvd_dvd_int @ B7 @ B )
          & ( dvd_dvd_int @ C4 @ C ) ) ) ).

% division_decomp
thf(fact_4028_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4029_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4030_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4031_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4032_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4033_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4034_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4035_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4036_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4037_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4038_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4039_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4040_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4041_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4042_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4043_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4044_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4045_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4046_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4047_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4048_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4049_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4050_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4051_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4052_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4053_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4054_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4055_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4056_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4057_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4058_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4059_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4060_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4061_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4062_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4063_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B2: code_integer,A3: code_integer] :
        ? [K2: code_integer] :
          ( A3
          = ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4064_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B2: real,A3: real] :
        ? [K2: real] :
          ( A3
          = ( times_times_real @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4065_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B2: rat,A3: rat] :
        ? [K2: rat] :
          ( A3
          = ( times_times_rat @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4066_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A3: nat] :
        ? [K2: nat] :
          ( A3
          = ( times_times_nat @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4067_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A3: int] :
        ? [K2: int] :
          ( A3
          = ( times_times_int @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4068_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_4069_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_4070_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_4071_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_4072_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_4073_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4074_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4075_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K3: rat] :
            ( A
           != ( times_times_rat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4076_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4077_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4078_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4079_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4080_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4081_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_4082_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_4083_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_4084_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_4085_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_4086_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_4087_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_4088_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_4089_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_4090_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4091_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4092_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4093_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4094_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4095_dvd__add__right__iff,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) )
        = ( dvd_dvd_complex @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4096_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4097_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4098_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4099_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4100_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4101_dvd__add__left__iff,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ C )
     => ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) )
        = ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4102_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4103_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4104_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4105_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4106_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4107_dvd__add,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( ( dvd_dvd_complex @ A @ C )
       => ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4108_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4109_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4110_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( dvd_dvd_Code_integer @ X @ Z )
       => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4111_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4112_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4113_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4114_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_4115_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_4116_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_4117_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4118_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4119_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4120_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4121_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4122_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4123_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4124_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4125_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4126_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4127_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4128_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4129_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4130_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4131_dvd__power__same,axiom,
    ! [X: real,Y: real,N2: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4132_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N2: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4133_dvd__power__same,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4134_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4135_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4136_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4137_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X22: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4138_old_Oprod_Ocase,axiom,
    ! [F: int > int > int,X1: int,X22: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4139_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4140_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4141_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4142_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4143_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4144_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4145_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4146_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4147_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4148_dvd__mod,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_4149_dvd__mod,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_4150_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N2: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N2 )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_4151_signed__take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_4152_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_4153_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_4154_signed__take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_4155_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4156_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4157_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4158_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4159_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4160_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4161_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4162_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4163_case__prodE2,axiom,
    ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
      ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4164_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4165_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4166_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4167_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4168_case__prod__eta,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int
        @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4169_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4170_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4171_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4172_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4173_cond__case__prod__eta,axiom,
    ! [F: int > int > int,G: product_prod_int_int > int] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc8211389475949308722nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4174_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4175_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4176_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4177_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4178_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4179_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4180_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_4181_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_4182_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_4183_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4184_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4185_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4186_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4187_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4188_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4189_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4190_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4191_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4192_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4193_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4194_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4195_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4196_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4197_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4198_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X5: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4199_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4200_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4201_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4202_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4203_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4204_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4205_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( ( divide_divide_real @ A @ B )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4206_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4207_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4208_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4209_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4210_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4211_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4212_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4213_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4214_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4215_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4216_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4217_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4218_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4219_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4220_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4221_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4222_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4223_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4224_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4225_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4226_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4227_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_4228_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_4229_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_4230_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4231_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4232_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4233_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4234_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4235_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4236_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4237_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4238_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4239_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4240_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4241_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4242_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4243_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4244_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4245_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4246_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4247_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4248_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4249_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4250_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4251_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4252_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4253_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4254_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_4255_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_4256_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_4257_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4258_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4259_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4260_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4261_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4262_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4263_div__power,axiom,
    ! [B: code_integer,A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N2 )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_4264_div__power,axiom,
    ! [B: nat,A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_4265_div__power,axiom,
    ! [B: int,A: int,N2: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
        = ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_4266_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4267_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4268_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4269_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A3 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4270_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A3: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A3 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4271_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4272_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4273_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4274_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4275_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4276_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4277_dvd__power__le,axiom,
    ! [X: real,Y: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4278_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4279_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4280_power__le__dvd,axiom,
    ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4281_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4282_power__le__dvd,axiom,
    ! [A: real,N2: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4283_power__le__dvd,axiom,
    ! [A: complex,N2: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4284_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_4285_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_4286_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_4287_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_4288_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_4289_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_4290_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4291_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4292_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4293_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4294_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4295_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_4296_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_4297_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_4298_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_4299_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_4300_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_4301_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_4302_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_4303_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_4304_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_4305_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_4306_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_4307_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_4308_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_4309_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4310_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4311_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4312_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
      = ( ? [X2: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4313_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X2: complex] : ( P @ ( times_times_complex @ L @ X2 ) ) )
      = ( ? [X2: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4314_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X2: real] : ( P @ ( times_times_real @ L @ X2 ) ) )
      = ( ? [X2: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X2 @ zero_zero_real ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4315_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4316_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4317_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4318_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4319_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4320_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4321_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4322_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4323_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4324_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4325_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4326_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4327_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4328_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4329_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4330_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4331_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4332_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4333_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_4334_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_4335_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_4336_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X5: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4337_inf__period_I4_J,axiom,
    ! [D: complex,D4: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X5: complex,K4: complex] :
          ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4338_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X5: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4339_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X5: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4340_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4341_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X5: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4342_inf__period_I3_J,axiom,
    ! [D: complex,D4: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X5: complex,K4: complex] :
          ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X5 @ T ) )
          = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4343_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X5: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4344_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X5: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4345_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4346_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4347_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4348_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4349_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4350_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4351_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4352_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_4353_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_4354_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_4355_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4356_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4357_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4358_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4359_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4360_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4361_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_4362_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_4363_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_4364_is__unit__power__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4365_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4366_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4367_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_4368_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_4369_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_4370_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_4371_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_4372_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_4373_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_4374_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_4375_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X2: nat] :
          ( collect_nat
          @ ^ [N3: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_4376_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q3 )
          = ( modulo_modulo_nat @ N2 @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4377_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_4378_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_4379_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_4380_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_4381_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4382_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4383_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4384_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4385_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4386_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4387_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B6: code_integer] :
              ( ( B6 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B6 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B6 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B6 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B6 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B6 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4388_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B6: nat] :
              ( ( B6 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B6 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B6 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B6 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B6 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B6 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4389_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B6: int] :
              ( ( B6 != zero_zero_int )
             => ( ( dvd_dvd_int @ B6 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B6 )
                 => ( ( ( divide_divide_int @ one_one_int @ B6 )
                      = A )
                   => ( ( ( times_times_int @ A @ B6 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B6 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4390_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B6: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_4391_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B6: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_4392_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B6: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_4393_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4394_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4395_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4396_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4397_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4398_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4399_bit__eq__rec,axiom,
    ( ( ^ [Y5: code_integer,Z5: code_integer] : Y5 = Z5 )
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4400_bit__eq__rec,axiom,
    ( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4401_bit__eq__rec,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [A3: int,B2: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4402_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N2: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4403_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4404_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4405_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_4406_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_4407_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_4408_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_4409_dvd__power,axiom,
    ! [N2: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_4410_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_4411_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_4412_dvd__power,axiom,
    ! [N2: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_4413_dvd__power,axiom,
    ! [N2: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_4414_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_4415_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_4416_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_4417_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_4418_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_4419_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_4420_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_4421_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_4422_dvd__minus__add,axiom,
    ! [Q3: nat,N2: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N2 )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q3 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4423_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_4424_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N2 )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4425_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_4426_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4427_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4428_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4429_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4430_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4431_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4432_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4433_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4434_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4435_power__mono__odd,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_4436_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_4437_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_4438_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% odd_pos
thf(fact_4439_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_4440_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_4441_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4442_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4443_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4444_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4445_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4446_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4447_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4448_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4449_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4450_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_4451_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B6: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4452_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B6: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_4453_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B6: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_4454_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_4455_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_4456_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_4457_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_4458_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_4459_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_4460_zero__le__even__power,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_4461_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_4462_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_4463_zero__le__odd__power,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4464_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4465_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4466_zero__le__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4467_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4468_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4469_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_4470_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_4471_zero__less__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4472_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4473_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4474_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_4475_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4476_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4477_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4478_power__le__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4479_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4480_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4481_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4482_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4483_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4484_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_4485_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_4486_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_4487_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_4488_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_4489_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_4490_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_4491_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4492_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4493_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4494_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4495_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4496_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4497_infinite__growing,axiom,
    ! [X8: set_real] :
      ( ( X8 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X8 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X8 )
                & ( ord_less_real @ X3 @ Xa ) ) )
       => ~ ( finite_finite_real @ X8 ) ) ) ).

% infinite_growing
thf(fact_4498_infinite__growing,axiom,
    ! [X8: set_rat] :
      ( ( X8 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X8 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X8 )
                & ( ord_less_rat @ X3 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X8 ) ) ) ).

% infinite_growing
thf(fact_4499_infinite__growing,axiom,
    ! [X8: set_num] :
      ( ( X8 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X8 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X8 )
                & ( ord_less_num @ X3 @ Xa ) ) )
       => ~ ( finite_finite_num @ X8 ) ) ) ).

% infinite_growing
thf(fact_4500_infinite__growing,axiom,
    ! [X8: set_nat] :
      ( ( X8 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X8 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X8 )
                & ( ord_less_nat @ X3 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X8 ) ) ) ).

% infinite_growing
thf(fact_4501_infinite__growing,axiom,
    ! [X8: set_int] :
      ( ( X8 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X8 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X8 )
                & ( ord_less_int @ X3 @ Xa ) ) )
       => ~ ( finite_finite_int @ X8 ) ) ) ).

% infinite_growing
thf(fact_4502_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4503_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4504_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4505_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4506_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4507_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_4508_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_4509_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_4510_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4511_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4512_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_4513_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4514_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4515_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4516_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_4517_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_4518_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_4519_intind,axiom,
    ! [I: nat,N2: nat,P: int > $o,X: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_int @ ( replicate_int @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_4520_intind,axiom,
    ! [I: nat,N2: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_4521_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_4522_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4523_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4524_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4525_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4526_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4527_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_4528_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4529_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4530_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4531_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4532_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
      ( ! [A5: code_integer,B6: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A5 @ B6 ) )
         => ( C @ A5 @ B6 ) )
     => ( produc7828578312038201481er_o_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_4533_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: num > num > $o] :
      ( ! [A5: num,B6: num] :
          ( ( P2
            = ( product_Pair_num_num @ A5 @ B6 ) )
         => ( C @ A5 @ B6 ) )
     => ( produc5703948589228662326_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_4534_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: nat > num > $o] :
      ( ! [A5: nat,B6: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A5 @ B6 ) )
         => ( C @ A5 @ B6 ) )
     => ( produc4927758841916487424_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_4535_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A5: nat,B6: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B6 ) )
         => ( C @ A5 @ B6 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_4536_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: int > int > $o] :
      ( ! [A5: int,B6: int] :
          ( ( P2
            = ( product_Pair_int_int @ A5 @ B6 ) )
         => ( C @ A5 @ B6 ) )
     => ( produc4947309494688390418_int_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_4537_case__prodI,axiom,
    ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
      ( ( F @ A @ B )
     => ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ).

% case_prodI
thf(fact_4538_case__prodI,axiom,
    ! [F: num > num > $o,A: num,B: num] :
      ( ( F @ A @ B )
     => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).

% case_prodI
thf(fact_4539_case__prodI,axiom,
    ! [F: nat > num > $o,A: nat,B: num] :
      ( ( F @ A @ B )
     => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).

% case_prodI
thf(fact_4540_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_4541_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( F @ A @ B )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_4542_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
      ( ! [A5: code_integer,B6: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A5 @ B6 ) )
         => ( member_complex @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4543_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
      ( ! [A5: code_integer,B6: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A5 @ B6 ) )
         => ( member_real @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4544_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
      ( ! [A5: code_integer,B6: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A5 @ B6 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4545_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
      ( ! [A5: code_integer,B6: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A5 @ B6 ) )
         => ( member_int @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4546_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: num > num > set_complex] :
      ( ! [A5: num,B6: num] :
          ( ( P2
            = ( product_Pair_num_num @ A5 @ B6 ) )
         => ( member_complex @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4547_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: num > num > set_real] :
      ( ! [A5: num,B6: num] :
          ( ( P2
            = ( product_Pair_num_num @ A5 @ B6 ) )
         => ( member_real @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4548_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: num > num > set_nat] :
      ( ! [A5: num,B6: num] :
          ( ( P2
            = ( product_Pair_num_num @ A5 @ B6 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4549_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: num > num > set_int] :
      ( ! [A5: num,B6: num] :
          ( ( P2
            = ( product_Pair_num_num @ A5 @ B6 ) )
         => ( member_int @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4550_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: complex,C: nat > num > set_complex] :
      ( ! [A5: nat,B6: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A5 @ B6 ) )
         => ( member_complex @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4551_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: nat > num > set_real] :
      ( ! [A5: nat,B6: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A5 @ B6 ) )
         => ( member_real @ Z @ ( C @ A5 @ B6 ) ) )
     => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_4552_mem__case__prodI,axiom,
    ! [Z: complex,C: code_integer > $o > set_complex,A: code_integer,B: $o] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4553_mem__case__prodI,axiom,
    ! [Z: real,C: code_integer > $o > set_real,A: code_integer,B: $o] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4554_mem__case__prodI,axiom,
    ! [Z: nat,C: code_integer > $o > set_nat,A: code_integer,B: $o] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4555_mem__case__prodI,axiom,
    ! [Z: int,C: code_integer > $o > set_int,A: code_integer,B: $o] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4556_mem__case__prodI,axiom,
    ! [Z: complex,C: num > num > set_complex,A: num,B: num] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4557_mem__case__prodI,axiom,
    ! [Z: real,C: num > num > set_real,A: num,B: num] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4558_mem__case__prodI,axiom,
    ! [Z: nat,C: num > num > set_nat,A: num,B: num] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4559_mem__case__prodI,axiom,
    ! [Z: int,C: num > num > set_int,A: num,B: num] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4560_mem__case__prodI,axiom,
    ! [Z: complex,C: nat > num > set_complex,A: nat,B: num] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4561_mem__case__prodI,axiom,
    ! [Z: real,C: nat > num > set_real,A: nat,B: num] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4562_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A5: nat,B6: nat] :
          ( ( ( product_Pair_nat_nat @ A5 @ B6 )
            = P2 )
         => ( C @ A5 @ B6 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P2 @ X ) ) ).

% case_prodI2'
thf(fact_4563_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_4564_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4565_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4566_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4567_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_4568_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4569_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4570_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4571_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4572_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_4573_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_4574_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4575_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4576_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4577_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4578_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_4579_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_4580_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4581_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_4582_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_4583_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4584_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4585_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4586_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_4587_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4588_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4589_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_4590_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_4591_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_4592_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4593_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4594_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4595_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4596_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4597_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4598_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4599_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4600_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_4601_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4602_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4603_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4604_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4605_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_4606_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4607_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4608_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4609_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4610_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_4611_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4612_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4613_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4614_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4615_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_4616_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4617_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4618_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4619_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4620_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_4621_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4622_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4623_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4624_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4625_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_4626_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_4627_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_4628_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_4629_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_4630_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_4631_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4632_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4633_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4634_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4635_div__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ A @ B ) ) ).

% div_minus_minus
thf(fact_4636_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_4637_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4638_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4639_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4640_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4641_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4642_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4643_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4644_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4645_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4646_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4647_mod__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4648_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4649_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4650_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4651_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4652_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4653_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_4654_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_4655_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_4656_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_4657_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_4658_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_4659_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4660_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4661_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4662_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4663_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4664_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4665_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_4666_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4667_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4668_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4669_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4670_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4671_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_4672_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_4673_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4674_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4675_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4676_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_4677_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4678_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4679_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4680_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4681_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4682_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4683_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X )
        = ( replicate_VEBT_VEBT @ N2 @ Y ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_4684_length__replicate,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_4685_length__replicate,axiom,
    ! [N2: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_4686_length__replicate,axiom,
    ! [N2: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_4687_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_4688_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_4689_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          | Q ) )
      = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_4690_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_4691_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_4692_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_4693_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_4694_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4695_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4696_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4697_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_4698_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_4699_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_4700_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_4701_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_4702_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4703_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4704_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4705_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_4706_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_4707_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_4708_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_4709_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_4710_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_4711_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_4712_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_4713_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_4714_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_4715_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_4716_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_4717_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_4718_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_4719_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4720_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4721_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4722_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4723_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4724_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4725_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_4726_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_4727_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4728_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4729_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4730_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_4731_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_4732_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_4733_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4734_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4735_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4736_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4737_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4738_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4739_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4740_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_4741_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_4742_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4743_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4744_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4745_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4746_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4747_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_4748_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_4749_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_4750_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_4751_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_4752_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_4753_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_4754_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_4755_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_4756_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_4757_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_4758_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4759_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4760_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4761_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4762_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_4763_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4764_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4765_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4766_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4767_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_4768_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_4769_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_4770_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_4771_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_4772_minus__mod__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_4773_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_4774_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4775_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4776_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4777_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4778_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4779_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4780_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4781_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4782_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4783_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4784_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4785_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4786_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4787_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4788_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4789_Suc__0__mod__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N2
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_4790_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4791_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4792_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4793_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4794_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4795_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4796_in__set__replicate,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4797_in__set__replicate,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4798_in__set__replicate,axiom,
    ! [X: set_nat,N2: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4799_in__set__replicate,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4800_in__set__replicate,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4801_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4802_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_4803_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_4804_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_4805_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_4806_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_4807_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_4808_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4809_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4810_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4811_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4812_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4813_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_4814_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_4815_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_4816_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_4817_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_4818_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_4819_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = zero_zero_real ) ).

% add_neg_numeral_special(8)
thf(fact_4820_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = zero_zero_complex ) ).

% add_neg_numeral_special(8)
thf(fact_4821_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_4822_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_4823_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_4824_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% add_neg_numeral_special(7)
thf(fact_4825_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% add_neg_numeral_special(7)
thf(fact_4826_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_4827_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_4828_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_4829_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% diff_numeral_special(12)
thf(fact_4830_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% diff_numeral_special(12)
thf(fact_4831_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_4832_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_4833_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4834_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4835_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4836_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4837_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4838_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4839_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4840_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4841_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4842_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4843_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4844_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4845_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4846_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4847_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4848_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_4849_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_4850_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_4851_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_4852_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_4853_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_4854_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_4855_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_4856_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_4857_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_4858_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_4859_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_4860_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_4861_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_4862_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_4863_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_4864_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_4865_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_4866_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_4867_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4868_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4869_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4870_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4871_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4872_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4873_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4874_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4875_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4876_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4877_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4878_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4879_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4880_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4881_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4882_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4883_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4884_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4885_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4886_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4887_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4888_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4889_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4890_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4891_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4892_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4893_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4894_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4895_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4896_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4897_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4898_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4899_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4900_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4901_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4902_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4903_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4904_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4905_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4906_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4907_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4908_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4909_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4910_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4911_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4912_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_4913_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_4914_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_4915_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_4916_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_4917_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4918_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4919_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4920_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4921_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_4922_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% diff_numeral_Suc
thf(fact_4923_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4924_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4925_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4926_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4927_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% max_numeral_Suc
thf(fact_4928_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_4929_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_4930_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_4931_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_4932_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_4933_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_4934_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_4935_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_4936_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_4937_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_4938_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4939_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4940_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4941_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4942_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4943_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4944_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4945_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4946_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4947_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4948_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4949_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4950_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4951_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4952_power2__minus,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4953_power2__minus,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4954_power2__minus,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4955_power2__minus,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4956_power2__minus,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4957_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4958_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4959_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4960_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4961_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4962_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4963_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4964_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4965_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4966_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4967_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4968_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4969_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4970_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4971_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4972_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4973_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4974_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4975_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_4976_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_4977_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_4978_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_4979_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_4980_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_4981_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4982_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4983_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4984_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4985_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4986_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_4987_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_4988_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_4989_power__minus__odd,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4990_power__minus__odd,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4991_power__minus__odd,axiom,
    ! [N2: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4992_power__minus__odd,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4993_power__minus__odd,axiom,
    ! [N2: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4994_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
        = ( power_power_int @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4995_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
        = ( power_power_real @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4996_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
        = ( power_power_complex @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4997_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4998_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
        = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4999_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5000_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5001_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5002_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5003_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5004_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5005_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5006_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5007_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5008_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_5009_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5010_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5011_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5012_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5013_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5014_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_5015_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_5016_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_5017_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_5018_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_5019_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_5020_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_5021_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_5022_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_5023_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_5024_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_5025_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_5026_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_5027_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_5028_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_5029_signed__take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5030_signed__take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5031_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_5032_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_5033_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5034_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5035_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5036_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5037_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5038_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5039_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_5040_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5041_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5042_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5043_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_5044_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_5045_signed__take__bit__minus,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_5046_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = ( zero_n2687167440665602831ol_nat @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_5047_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = ( zero_n2684676970156552555ol_int @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_5048_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = ( zero_n356916108424825756nteger @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_5049_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_5050_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_5051_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_5052_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_5053_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_5054_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5055_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5056_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5057_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5058_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5059_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_5060_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_5061_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_5062_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_5063_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_5064_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_5065_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_5066_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_5067_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_5068_mem__case__prodE,axiom,
    ! [Z: complex,C: code_integer > $o > set_complex,P2: produc6271795597528267376eger_o] :
      ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P2 ) )
     => ~ ! [X3: code_integer,Y3: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X3 @ Y3 ) )
           => ~ ( member_complex @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5069_mem__case__prodE,axiom,
    ! [Z: real,C: code_integer > $o > set_real,P2: produc6271795597528267376eger_o] :
      ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P2 ) )
     => ~ ! [X3: code_integer,Y3: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5070_mem__case__prodE,axiom,
    ! [Z: nat,C: code_integer > $o > set_nat,P2: produc6271795597528267376eger_o] :
      ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P2 ) )
     => ~ ! [X3: code_integer,Y3: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5071_mem__case__prodE,axiom,
    ! [Z: int,C: code_integer > $o > set_int,P2: produc6271795597528267376eger_o] :
      ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P2 ) )
     => ~ ! [X3: code_integer,Y3: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5072_mem__case__prodE,axiom,
    ! [Z: complex,C: num > num > set_complex,P2: product_prod_num_num] :
      ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_complex @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5073_mem__case__prodE,axiom,
    ! [Z: real,C: num > num > set_real,P2: product_prod_num_num] :
      ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5074_mem__case__prodE,axiom,
    ! [Z: nat,C: num > num > set_nat,P2: product_prod_num_num] :
      ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5075_mem__case__prodE,axiom,
    ! [Z: int,C: num > num > set_int,P2: product_prod_num_num] :
      ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5076_mem__case__prodE,axiom,
    ! [Z: complex,C: nat > num > set_complex,P2: product_prod_nat_num] :
      ( ( member_complex @ Z @ ( produc6231982587499038204omplex @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( member_complex @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5077_mem__case__prodE,axiom,
    ! [Z: real,C: nat > num > set_real,P2: product_prod_nat_num] :
      ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_5078_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5079_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5080_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5081_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_5082_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5083_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5084_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5085_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_5086_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_5087_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_5088_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_5089_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_5090_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_5091_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5092_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5093_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5094_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_5095_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_5096_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_5097_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_5098_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_5099_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5100_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5101_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5102_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5103_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5104_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5105_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5106_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5107_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5108_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5109_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5110_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5111_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5112_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_5113_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5114_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5115_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5116_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5117_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_5118_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_5119_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_5120_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_5121_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_5122_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_5123_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_5124_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_5125_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_5126_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_5127_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_5128_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5129_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5130_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5131_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5132_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_5133_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5134_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5135_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5136_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5137_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_5138_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5139_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5140_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5141_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5142_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_5143_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5144_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5145_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5146_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5147_minus__diff__minus,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5148_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5149_minus__diff__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5150_minus__diff__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5151_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5152_div__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5153_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5154_minus__divide__right,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_divide_right
thf(fact_5155_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_5156_minus__divide__right,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_divide_right
thf(fact_5157_minus__divide__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( divide_divide_real @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5158_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5159_minus__divide__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( divide_divide_rat @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5160_minus__divide__left,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5161_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5162_minus__divide__left,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5163_mod__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_5164_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_5165_mod__minus__cong,axiom,
    ! [A: int,B: int,A4: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A4 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5166_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A4 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5167_mod__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5168_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5169_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_5170_case__prodE,axiom,
    ! [C: code_integer > $o > $o,P2: produc6271795597528267376eger_o] :
      ( ( produc7828578312038201481er_o_o @ C @ P2 )
     => ~ ! [X3: code_integer,Y3: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5171_case__prodE,axiom,
    ! [C: num > num > $o,P2: product_prod_num_num] :
      ( ( produc5703948589228662326_num_o @ C @ P2 )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5172_case__prodE,axiom,
    ! [C: nat > num > $o,P2: product_prod_nat_num] :
      ( ( produc4927758841916487424_num_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5173_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5174_case__prodE,axiom,
    ! [C: int > int > $o,P2: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P2 )
     => ~ ! [X3: int,Y3: int] :
            ( ( P2
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_5175_case__prodD,axiom,
    ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
      ( ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5176_case__prodD,axiom,
    ! [F: num > num > $o,A: num,B: num] :
      ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5177_case__prodD,axiom,
    ! [F: nat > num > $o,A: nat,B: num] :
      ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5178_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5179_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_5180_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P2 @ Z )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 @ Z ) ) ) ).

% case_prodE'
thf(fact_5181_case__prodD_H,axiom,
    ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_5182_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5183_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5184_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5185_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5186_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5187_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_5188_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_5189_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_5190_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_5191_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_5192_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_5193_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_5194_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_5195_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_5196_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_5197_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_5198_split__of__bool,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ( P2
         => ( P @ one_one_complex ) )
        & ( ~ P2
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_5199_split__of__bool,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ( P2
         => ( P @ one_one_real ) )
        & ( ~ P2
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_5200_split__of__bool,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_rat ) )
        & ( ~ P2
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_5201_split__of__bool,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_nat ) )
        & ( ~ P2
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_5202_split__of__bool,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ( P2
         => ( P @ one_one_int ) )
        & ( ~ P2
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_5203_split__of__bool,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ( P2
         => ( P @ one_one_Code_integer ) )
        & ( ~ P2
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_5204_split__of__bool__asm,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_complex ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5205_split__of__bool__asm,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_real ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5206_split__of__bool__asm,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_rat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5207_split__of__bool__asm,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_nat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5208_split__of__bool__asm,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_int ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5209_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P2
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5210_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5211_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5212_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5213_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5214_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5215_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5216_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5217_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5218_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5219_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5220_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5221_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5222_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5223_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5224_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5225_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5226_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5227_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5228_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5229_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5230_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5231_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% le_minus_one_simps(2)
thf(fact_5232_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_5233_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_5234_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_5235_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(4)
thf(fact_5236_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_5237_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_5238_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_5239_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5240_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5241_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5242_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5243_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5244_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_5245_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5246_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5247_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5248_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5249_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_5250_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5251_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5252_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5253_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5254_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_5255_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5256_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5257_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5258_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5259_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_5260_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5261_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5262_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5263_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5264_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_5265_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5266_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5267_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5268_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5269_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_5270_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(4)
thf(fact_5271_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_5272_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_5273_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_5274_less__minus__one__simps_I2_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% less_minus_one_simps(2)
thf(fact_5275_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_5276_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_5277_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5278_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
      = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5279_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
      = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5280_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5281_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5282_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5283_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5284_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5285_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5286_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5287_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5288_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ N2 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5289_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ N2 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5290_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ N2 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5291_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ N2 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5292_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ N2 )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5293_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5294_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5295_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5296_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5297_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5298_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_5299_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_5300_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_5301_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5302_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5303_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_5304_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5305_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5306_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5307_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5308_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_5309_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5310_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5311_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5312_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5313_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_5314_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5315_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5316_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5317_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5318_replicate__eqI,axiom,
    ! [Xs2: list_complex,N2: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N2 )
     => ( ! [Y3: complex] :
            ( ( member_complex @ Y3 @ ( set_complex2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_complex @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5319_replicate__eqI,axiom,
    ! [Xs2: list_real,N2: nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N2 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_real @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5320_replicate__eqI,axiom,
    ! [Xs2: list_set_nat,N2: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs2 )
        = N2 )
     => ( ! [Y3: set_nat] :
            ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_set_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5321_replicate__eqI,axiom,
    ! [Xs2: list_nat,N2: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N2 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5322_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5323_replicate__eqI,axiom,
    ! [Xs2: list_o,N2: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N2 )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_o @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5324_replicate__eqI,axiom,
    ! [Xs2: list_int,N2: nat,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N2 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_int @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5325_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5326_replicate__length__same,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5327_replicate__length__same,axiom,
    ! [Xs2: list_int,X: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5328_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5329_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5330_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5331_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5332_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5333_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5334_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5335_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5336_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5337_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5338_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_5339_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_5340_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_5341_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_5342_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_5343_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_5344_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).

% minus_real_def
thf(fact_5345_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5346_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5347_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5348_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5349_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_5350_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5351_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_5352_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_5353_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_5354_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_5355_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_5356_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5357_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5358_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5359_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5360_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5361_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% le_minus_one_simps(1)
thf(fact_5362_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5363_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_5364_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_5365_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(3)
thf(fact_5366_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5367_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_5368_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_5369_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_5370_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(3)
thf(fact_5371_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_5372_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5373_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_5374_less__minus__one__simps_I1_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% less_minus_one_simps(1)
thf(fact_5375_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_5376_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5377_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_5378_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5379_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_5380_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_5381_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_5382_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5383_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_5384_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_5385_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_5386_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_5387_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_5388_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_5389_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_5390_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5391_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_5392_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_5393_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5394_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5395_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5396_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5397_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_5398_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_5399_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_5400_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5401_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_5402_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_5403_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_5404_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5405_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_5406_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_5407_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_5408_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5409_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5410_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5411_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5412_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5413_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5414_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5415_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5416_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5417_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5418_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5419_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5420_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5421_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5422_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5423_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5424_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5425_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5426_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5427_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5428_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5429_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5430_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5431_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5432_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5433_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5434_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5435_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5436_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5437_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5438_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5439_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5440_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5441_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5442_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5443_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_5444_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_5445_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5446_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5447_power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5448_power__minus,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5449_power__minus,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5450_power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5451_power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5452_power__minus__Bit0,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5453_power__minus__Bit0,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5454_power__minus__Bit0,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5455_power__minus__Bit0,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5456_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5457_power__minus__Bit1,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5458_power__minus__Bit1,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5459_power__minus__Bit1,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5460_power__minus__Bit1,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5461_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5462_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_5463_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_5464_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_5465_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_5466_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_5467_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_5468_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_5469_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5470_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5471_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5472_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5473_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5474_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5475_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5476_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5477_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5478_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5479_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5480_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5481_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5482_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5483_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5484_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5485_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5486_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5487_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5488_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5489_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5490_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5491_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5492_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5493_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5494_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5495_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5496_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5497_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5498_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5499_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5500_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5501_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5502_even__minus,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5503_even__minus,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5504_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5505_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5506_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5507_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5508_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5509_uminus__power__if,axiom,
    ! [N2: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
          = ( power_power_int @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5510_uminus__power__if,axiom,
    ! [N2: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
          = ( power_power_real @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5511_uminus__power__if,axiom,
    ! [N2: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
          = ( power_power_complex @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5512_uminus__power__if,axiom,
    ! [N2: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
          = ( power_power_rat @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5513_uminus__power__if,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
          = ( power_8256067586552552935nteger @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5514_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,N2: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_5515_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_5516_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] :
      ( ( zero_n2687167440665602831ol_nat
        @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5517_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] :
      ( ( zero_n2684676970156552555ol_int
        @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5518_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] :
      ( ( zero_n356916108424825756nteger
        @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5519_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5520_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5521_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5522_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5523_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5524_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5525_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5526_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5527_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5528_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5529_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5530_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5531_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5532_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5533_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5534_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5535_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5536_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5537_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5538_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5539_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5540_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_5541_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_5542_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_5543_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_5544_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_5545_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5546_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5547_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5548_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5549_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5550_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_5551_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_5552_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_5553_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_5554_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5555_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_5556_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_5557_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_5558_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5559_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: nat,B6: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B6 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5560_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: int,B6: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B6 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5561_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A5: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_integer,B6: $o] :
            ( ( P @ A5 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B6 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5562_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5563_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5564_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5565_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5566_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_5567_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5568_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5569_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_5570_minus__power__mult__self,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5571_minus__power__mult__self,axiom,
    ! [A: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5572_minus__power__mult__self,axiom,
    ! [A: complex,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5573_minus__power__mult__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5574_minus__power__mult__self,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5575_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_ri631733984087533419it_int @ N2 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_5576_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_5577_minus__1__div__exp__eq__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_5578_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_5579_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5580_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5581_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5582_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N3 ) @ ( modulo_modulo_nat @ M5 @ N3 ) ) ) ) ).

% divmod_nat_def
thf(fact_5583_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_5584_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_5585_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_5586_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_5587_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_5588_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K3: int] :
              ( ( P @ K3 )
             => ( ( K3 != zero_zero_int )
               => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K3: int] :
                ( ( P @ K3 )
               => ( ( K3
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_5589_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_5590_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5591_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5592_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5593_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_5594_one__div__minus__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_5595_minus__one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_5596_numeral__div__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_5597_minus__numeral__div__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_5598_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5599_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5600_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5601_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5602_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5603_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_5604_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K2: int] :
          ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_5605_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K2: int] :
          ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_5606_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K2: int] :
          ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_5607_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K2: int] :
          ( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_5608_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K2: int] :
          ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_5609_split__part,axiom,
    ! [P: $o,Q: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A3: int,B2: int] :
            ( P
            & ( Q @ A3 @ B2 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P
            & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_5610_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_5611_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_5612_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_5613_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_5614_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5615_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5616_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5617_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5618_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_5619_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_5620_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_5621_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_5622_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_5623_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_5624_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_5625_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5626_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5627_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5628_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = one_one_complex )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5629_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5630_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5631_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5632_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_5633_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_5634_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_5635_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_5636_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_5637_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_5638_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_5639_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_5640_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_add
thf(fact_5641_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_5642_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_5643_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_5644_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_5645_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_5646_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5647_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5648_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5649_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5650_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5651_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5652_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5653_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5654_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_5655_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_5656_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_5657_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_5658_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5659_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5660_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5661_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5662_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_5663_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_5664_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_5665_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_5666_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_5667_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5668_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5669_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5670_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5671_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5672_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5673_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5674_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5675_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5676_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5677_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5678_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5679_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5680_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5681_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5682_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5683_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5684_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5685_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5686_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5687_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5688_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5689_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5690_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5691_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5692_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5693_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5694_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5695_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5696_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5697_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5698_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5699_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5700_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5701_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5702_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5703_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5704_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5705_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5706_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5707_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5708_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5709_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5710_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5711_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5712_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5713_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5714_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5715_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5716_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5717_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5718_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5719_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5720_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5721_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5722_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5723_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5724_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5725_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5726_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5727_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5728_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5729_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5730_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5731_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5732_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5733_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5734_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5735_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5736_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5737_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5738_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5739_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5740_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5741_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5742_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5743_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5744_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5745_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5746_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5747_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5748_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5749_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_5750_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5751_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5752_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5753_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5754_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_5755_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5756_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5757_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5758_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5759_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5760_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5761_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5762_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5763_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5764_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5765_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5766_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5767_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5768_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5769_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5770_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5771_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu3: int,Uv3: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_5772_mult__of__int__commute,axiom,
    ! [X: int,Y: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5773_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5774_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5775_real__of__int__div4,axiom,
    ! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_5776_real__of__int__div,axiom,
    ! [D: int,N2: int] :
      ( ( dvd_dvd_int @ D @ N2 )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_5777_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5778_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5779_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5780_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_5781_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_5782_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_5783_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5784_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5785_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5786_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5787_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_5788_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N3: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_5789_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N3: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).

% int_less_real_le
thf(fact_5790_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_5791_real__of__int__div2,axiom,
    ! [N2: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_5792_real__of__int__div3,axiom,
    ! [N2: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_5793_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_5794_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5795_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5796_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_5797_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_5798_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_5799_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_5800_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5801_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5802_floor__exists1,axiom,
    ! [X: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_5803_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_5804_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_5805_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_5806_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z3: int] :
                ( ( ord_less_eq_int @ D2 @ Z6 )
                & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_5807_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z3: int] :
                ( ( ord_less_eq_int @ D2 @ Z3 )
                & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_5808_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_5809_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5810_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X )
            = ( ln_ln_real @ Y ) )
          = ( X = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_5811_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_5812_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_5813_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_iff
thf(fact_5814_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = zero_zero_real )
        = ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_5815_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_5816_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_5817_ln__less__self,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_less_self
thf(fact_5818_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_5819_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_5820_ln__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_5821_ln__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_gt_zero
thf(fact_5822_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_5823_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_5824_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_5825_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_5826_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = ( minus_minus_real @ X @ one_one_real ) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_5827_ln__div,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_5828_ln__2__less__1,axiom,
    ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).

% ln_2_less_1
thf(fact_5829_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_5830_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_5831_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_5832_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_5833_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N: nat] :
            ( ~ ( P @ N )
            & ( P @ ( suc @ N ) ) ) ) ) ).

% exists_least_lemma
thf(fact_5834_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5835_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5836_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_5837_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_5838_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5839_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5840_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_5841_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5842_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5843_round__unique,axiom,
    ! [X: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_5844_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_5845_tanh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( tanh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_5846_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_5847_of__int__round__gt,axiom,
    ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_gt
thf(fact_5848_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_5849_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_5850_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_5851_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_5852_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_5853_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_5854_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_5855_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_5856_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_5857_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_5858_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_5859_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_5860_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_5861_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_5862_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_5863_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_5864_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_5865_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_5866_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_5867_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_5868_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_5869_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_5870_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_5871_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_5872_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_5873_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_5874_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_numeral
thf(fact_5875_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_numeral
thf(fact_5876_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_numeral
thf(fact_5877_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_numeral
thf(fact_5878_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5879_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5880_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5881_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_5882_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_5883_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_5884_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_5885_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_5886_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_5887_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_5888_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_5889_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_5890_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_5891_abs__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_divide
thf(fact_5892_abs__divide,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_divide
thf(fact_5893_abs__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_divide
thf(fact_5894_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_5895_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_5896_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_5897_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_5898_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_5899_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_5900_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_5901_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_5902_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_5903_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5904_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5905_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5906_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_5907_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5908_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5909_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5910_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_5911_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_5912_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_5913_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_5914_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_5915_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% tanh_real_less_iff
thf(fact_5916_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% tanh_real_le_iff
thf(fact_5917_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_5918_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_5919_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_5920_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_5921_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_5922_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_5923_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_5924_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_5925_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5926_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5927_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5928_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_5929_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_5930_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_5931_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_5932_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_5933_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5934_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5935_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5936_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_neg_numeral
thf(fact_5937_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_5938_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_5939_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_5940_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_5941_abs__power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_5942_abs__power__minus,axiom,
    ! [A: real,N2: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_5943_abs__power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_5944_abs__power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_5945_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_5946_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% tanh_real_pos_iff
thf(fact_5947_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_5948_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_5949_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_5950_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_5951_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_5952_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_5953_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_5954_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_5955_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_5956_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_5957_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5958_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5959_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5960_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_5961_artanh__minus__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
        = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).

% artanh_minus_real
thf(fact_5962_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5963_zero__less__power__abs__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
      = ( ( A != zero_zero_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5964_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
      = ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5965_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
      = ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5966_power2__abs,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5967_power2__abs,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5968_power2__abs,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5969_power2__abs,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5970_abs__power2,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5971_abs__power2,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5972_abs__power2,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5973_abs__power2,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5974_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_5975_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_5976_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5977_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5978_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5979_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5980_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_5981_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_5982_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_5983_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_5984_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_5985_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5986_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5987_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5988_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_5989_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_5990_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_5991_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_5992_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_5993_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_5994_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_5995_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_5996_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_5997_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5998_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_5999_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_6000_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_6001_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6002_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6003_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6004_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6005_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6006_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6007_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6008_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6009_power__abs,axiom,
    ! [A: rat,N2: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6010_power__abs,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6011_power__abs,axiom,
    ! [A: real,N2: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N2 ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6012_power__abs,axiom,
    ! [A: int,N2: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N2 ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6013_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6014_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6015_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( ( abs_abs_rat @ L )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6016_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6017_round__diff__minimal,axiom,
    ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_6018_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_6019_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_6020_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_6021_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_6022_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_6023_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6024_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6025_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6026_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6027_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_6028_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_6029_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_6030_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_6031_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6032_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6033_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6034_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6035_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6036_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6037_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6038_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6039_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6040_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6041_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6042_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6043_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6044_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6045_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6046_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6047_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6048_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6049_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6050_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6051_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_6052_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_6053_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_6054_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_6055_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_6056_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_6057_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6058_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6059_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6060_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6061_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6062_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6063_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6064_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6065_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6066_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6067_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6068_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6069_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6070_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6071_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6072_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_6073_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_6074_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_6075_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_6076_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_6077_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_6078_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_6079_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_6080_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6081_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6082_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6083_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6084_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6085_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6086_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6087_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6088_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6089_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6090_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6091_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6092_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_6093_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6094_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_6095_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_6096_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_6097_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_6098_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_6099_zero__le__power__abs,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_6100_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_6101_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_6102_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_6103_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_6104_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_6105_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_6106_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_6107_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_6108_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_6109_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_6110_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_6111_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_6112_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_6113_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_6114_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6115_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6116_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6117_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6118_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6119_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6120_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6121_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6122_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6123_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6124_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6125_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6126_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6127_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6128_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6129_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6130_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_6131_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_real_def
thf(fact_6132_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y4 )
                  & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_6133_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_6134_tanh__real__gt__neg1,axiom,
    ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).

% tanh_real_gt_neg1
thf(fact_6135_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6136_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6137_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6138_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6139_of__int__leD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_6140_of__int__leD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_6141_of__int__leD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_6142_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_6143_of__int__lessD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_6144_of__int__lessD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_6145_of__int__lessD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_6146_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_6147_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_6148_of__int__round__abs__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_6149_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_6150_round__unique_H,axiom,
    ! [X: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_6151_round__unique_H,axiom,
    ! [X: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_6152_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6153_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6154_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6155_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6156_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_6157_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_6158_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_6159_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_6160_power__even__abs,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6161_power__even__abs,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 )
        = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6162_power__even__abs,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N2 )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6163_power__even__abs,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N2 )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6164_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6165_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6166_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6167_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6168_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
         => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6169_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6170_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
         => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6171_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6172_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_6173_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_6174_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_6175_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_6176_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_6177_abs__square__less__1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_6178_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_6179_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_6180_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6181_power__mono__even,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6182_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6183_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_6184_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6185_of__int__round__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_6186_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_6187_of__int__round__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_ge
thf(fact_6188_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_6189_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_6190_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_6191_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_6192_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_6193_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_6194_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_6195_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X2: int] : X2
          @ ( set_or1266510415728281911st_int @ M @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_6196_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).

% even_set_encode_iff
thf(fact_6197_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_6198_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_6199_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_6200_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_6201_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_6202_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_6203_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_6204_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_6205_abs__sum__abs,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( abs_abs_int
        @ ( groups4538972089207619220nt_int
          @ ^ [A3: int] : ( abs_abs_int @ ( F @ A3 ) )
          @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [A3: int] : ( abs_abs_int @ ( F @ A3 ) )
        @ A2 ) ) ).

% abs_sum_abs
thf(fact_6206_abs__sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ( abs_abs_real
        @ ( groups6591440286371151544t_real
          @ ^ [A3: nat] : ( abs_abs_real @ ( F @ A3 ) )
          @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [A3: nat] : ( abs_abs_real @ ( F @ A3 ) )
        @ A2 ) ) ).

% abs_sum_abs
thf(fact_6207_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_6208_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_6209_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N2 )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6210_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N2 )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6211_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_arctan_iff
thf(fact_6212_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_6213_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_6214_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_6215_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6216_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6217_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6218_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6219_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6220_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6221_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6222_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6223_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6224_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6225_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6226_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6227_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6228_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6229_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6230_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6231_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6232_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6233_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6234_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6235_sum__abs,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_6236_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_6237_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_6238_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_6239_of__int__sum,axiom,
    ! [F: complex > int,A2: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( ring_17405671764205052669omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6240_of__int__sum,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6241_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups8778361861064173332t_real
        @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6242_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6243_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6244_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_6245_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_6246_sum_Oswap,axiom,
    ! [G: int > int > int,B3: set_int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( groups4538972089207619220nt_int @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups4538972089207619220nt_int
        @ ^ [J3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [I3: int] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% sum.swap
thf(fact_6247_sum_Oswap,axiom,
    ! [G: complex > complex > complex,B3: set_complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [I3: complex] : ( groups7754918857620584856omplex @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups7754918857620584856omplex
        @ ^ [J3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [I3: complex] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% sum.swap
thf(fact_6248_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% sum.swap
thf(fact_6249_sum_Oswap,axiom,
    ! [G: nat > nat > real,B3: set_nat,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% sum.swap
thf(fact_6250_of__int__mask__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% of_int_mask_eq
thf(fact_6251_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% arctan_less_iff
thf(fact_6252_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_6253_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_6254_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% arctan_le_iff
thf(fact_6255_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_6256_sum__mono,axiom,
    ! [K5: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6257_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6258_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6259_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6260_sum__mono,axiom,
    ! [K5: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6261_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6262_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6263_sum__mono,axiom,
    ! [K5: set_complex,F: complex > int,G: complex > int] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6264_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6265_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6266_sum__distrib__left,axiom,
    ! [R2: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N3: int] : ( times_times_int @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_6267_sum__distrib__left,axiom,
    ! [R2: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( times_times_complex @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_6268_sum__distrib__left,axiom,
    ! [R2: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N3: nat] : ( times_times_nat @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_6269_sum__distrib__left,axiom,
    ! [R2: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( times_times_real @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_6270_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R2: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
      = ( groups4538972089207619220nt_int
        @ ^ [N3: int] : ( times_times_int @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_6271_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_6272_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R2: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
      = ( groups3542108847815614940at_nat
        @ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_6273_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_6274_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_6275_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_6276_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_6277_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_6278_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_6279_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_6280_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_6281_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_6282_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
        @ A2 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_6283_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
        @ A2 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_6284_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
        @ A2 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_6285_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_6286_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_6287_sum__negf,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( uminus_uminus_int @ ( F @ X2 ) )
        @ A2 )
      = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).

% sum_negf
thf(fact_6288_sum__negf,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( uminus1482373934393186551omplex @ ( F @ X2 ) )
        @ A2 )
      = ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A2 ) ) ) ).

% sum_negf
thf(fact_6289_sum__negf,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( uminus_uminus_real @ ( F @ X2 ) )
        @ A2 )
      = ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).

% sum_negf
thf(fact_6290_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_int,G: real > int > int,R: real > int > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups1932886352136224148al_int
            @ ^ [X2: real] :
                ( groups4538972089207619220nt_int @ ( G @ X2 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y2: int] :
                ( groups1932886352136224148al_int
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6291_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_int,G: nat > int > int,R: nat > int > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [X2: nat] :
                ( groups4538972089207619220nt_int @ ( G @ X2 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y2: int] :
                ( groups3539618377306564664at_int
                @ ^ [X2: nat] : ( G @ X2 @ Y2 )
                @ ( collect_nat
                  @ ^ [X2: nat] :
                      ( ( member_nat @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6292_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_int,G: complex > int > int,R: complex > int > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups5690904116761175830ex_int
            @ ^ [X2: complex] :
                ( groups4538972089207619220nt_int @ ( G @ X2 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups4538972089207619220nt_int
            @ ^ [Y2: int] :
                ( groups5690904116761175830ex_int
                @ ^ [X2: complex] : ( G @ X2 @ Y2 )
                @ ( collect_complex
                  @ ^ [X2: complex] :
                      ( ( member_complex @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6293_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_complex,G: real > complex > complex,R: real > complex > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X2: real] :
                ( groups7754918857620584856omplex @ ( G @ X2 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6294_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X2: nat] :
                ( groups7754918857620584856omplex @ ( G @ X2 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X2: nat] : ( G @ X2 @ Y2 )
                @ ( collect_nat
                  @ ^ [X2: nat] :
                      ( ( member_nat @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6295_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_complex,G: int > complex > complex,R: int > complex > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X2: int] :
                ( groups7754918857620584856omplex @ ( G @ X2 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y2: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X2: int] : ( G @ X2 @ Y2 )
                @ ( collect_int
                  @ ^ [X2: int] :
                      ( ( member_int @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6296_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X2: real] :
                ( groups3542108847815614940at_nat @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6297_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X2: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X2: complex] : ( G @ X2 @ Y2 )
                @ ( collect_complex
                  @ ^ [X2: complex] :
                      ( ( member_complex @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6298_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X2: int] :
                ( groups3542108847815614940at_nat @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y2: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X2: int] : ( G @ X2 @ Y2 )
                @ ( collect_int
                  @ ^ [X2: int] :
                      ( ( member_int @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6299_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: real > nat > real,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups8097168146408367636l_real
            @ ^ [X2: real] :
                ( groups6591440286371151544t_real @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups6591440286371151544t_real
            @ ^ [Y2: nat] :
                ( groups8097168146408367636l_real
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_6300_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_6301_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_6302_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6303_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6304_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6305_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6306_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6307_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6308_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6309_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6310_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6311_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6312_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6313_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6314_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6315_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6316_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6317_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6318_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6319_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6320_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6321_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6322_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6323_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6324_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6325_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6326_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6327_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6328_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6329_sum__mono__inv,axiom,
    ! [F: real > int,I5: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I5 )
        = ( groups1932886352136224148al_int @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6330_sum__mono__inv,axiom,
    ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I5 )
        = ( groups3539618377306564664at_int @ G @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6331_sum__mono__inv,axiom,
    ! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
      ( ( ( groups5690904116761175830ex_int @ F @ I5 )
        = ( groups5690904116761175830ex_int @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6332_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_6333_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_6334_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6335_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6336_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6337_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6338_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6339_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6340_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6341_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6342_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X2: complex] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6343_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X2: int] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_6344_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_complex ) ) ) )
        = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6345_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6346_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6347_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6348_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6349_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6350_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6351_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6352_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X2: nat] :
                  ( ( G @ X2 )
                  = zero_zero_complex ) ) ) )
        = ( groups2073611262835488442omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6353_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X2: nat] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6354_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_6355_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_6356_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6357_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6358_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6359_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6360_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6361_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6362_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6363_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: nat > rat,I: nat > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6364_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > rat,I: complex > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6365_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > rat,I: int > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6366_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6367_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6368_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6369_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6370_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6371_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6372_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6373_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6374_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6375_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6376_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6377_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6378_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6379_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6380_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6381_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6382_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6383_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
              & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6384_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6385_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6386_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6387_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6388_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6389_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6390_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6391_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6392_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6393_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6394_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6395_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6396_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6397_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6398_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6399_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6400_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6401_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6402_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6403_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6404_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6405_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6406_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6407_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6408_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6409_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6410_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6411_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6412_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6413_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > complex] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3049146728041665814omplex @ G @ A2 )
          = ( plus_plus_complex @ ( groups3049146728041665814omplex @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3049146728041665814omplex @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6414_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6415_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6416_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6417_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6418_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6419_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6420_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6421_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6422_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6423_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6424_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6425_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_6426_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6427_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6428_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6429_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6430_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6431_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6432_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6433_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6434_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6435_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6436_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B3: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6437_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B3: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6438_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B3: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6439_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B3: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6440_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B3: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B3 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6441_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B3: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6442_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B3: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6443_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B3: nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6444_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > nat,B3: nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6445_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > nat,B3: nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6446_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_6447_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% less_mask
thf(fact_6448_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6449_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6450_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6451_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6452_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6453_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6454_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6455_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6456_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6457_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6458_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B6: real] :
              ( ( member_real @ B6 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B6 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6459_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B6: complex] :
              ( ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B6 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6460_sum__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ! [B6: int] :
              ( ( member_int @ B6 @ ( minus_minus_set_int @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B6 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6461_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B6: real] :
              ( ( member_real @ B6 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B6 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6462_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B6: complex] :
              ( ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B6 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6463_sum__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ! [B6: int] :
              ( ( member_int @ B6 @ ( minus_minus_set_int @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B6 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6464_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B6: real] :
              ( ( member_real @ B6 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B6 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6465_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B6: complex] :
              ( ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B6 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6466_sum__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ! [B6: int] :
              ( ( member_int @ B6 @ ( minus_minus_set_int @ B3 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B6 ) ) )
         => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6467_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B6: real] :
              ( ( member_real @ B6 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B6 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_6468_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6469_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6470_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6471_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6472_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6473_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6474_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6475_sum__pos,axiom,
    ! [I5: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6476_sum__pos,axiom,
    ! [I5: set_int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6477_sum__pos,axiom,
    ! [I5: set_real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6478_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
        = ( ( abs_abs_int @ N2 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_6479_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6480_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6481_sum__strict__mono2,axiom,
    ! [B3: set_int,A2: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6482_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6483_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6484_sum__strict__mono2,axiom,
    ! [B3: set_int,A2: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6485_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6486_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6487_sum__strict__mono2,axiom,
    ! [B3: set_int,A2: set_int,B: int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6488_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > int] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6489_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_6490_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_6491_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_6492_Suc__mask__eq__exp,axiom,
    ! [N2: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_mask_eq_exp
thf(fact_6493_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_6494_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6495_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6496_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6497_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6498_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X @ I5 )
          = one_one_real )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6499_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ I5 )
          = one_one_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6500_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ I5 )
          = one_one_real )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6501_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I3: complex] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6502_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6503_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6504_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N2 )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_6505_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_6506_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_6507_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_6508_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_6509_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_6510_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_6511_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_6512_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_6513_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_6514_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_6515_crossproduct__eq,axiom,
    ! [W: complex,Y: complex,X: complex,Z: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X @ Z ) )
        = ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6516_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6517_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6518_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6519_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6520_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
       != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_6521_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_6522_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_6523_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_6524_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_6525_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_6526_mask__half__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_6527_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_6528_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_6529_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6530_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6531_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_6532_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_6533_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_6534_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6535_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6536_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_6537_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_6538_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( K2 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K2
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_6539_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_6540_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_6541_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_6542_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_6543_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_6544_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_6545_arctan__half,axiom,
    ( arctan
    = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_6546_and_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.right_idem
thf(fact_6547_and_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.right_idem
thf(fact_6548_and_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.left_idem
thf(fact_6549_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.left_idem
thf(fact_6550_and_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ A )
      = A ) ).

% and.idem
thf(fact_6551_and_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ A )
      = A ) ).

% and.idem
thf(fact_6552_or_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
      = ( bit_se1409905431419307370or_int @ A @ B ) ) ).

% or.right_idem
thf(fact_6553_or_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
      = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).

% or.right_idem
thf(fact_6554_or_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( bit_se1409905431419307370or_int @ A @ B ) ) ).

% or.left_idem
thf(fact_6555_or_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).

% or.left_idem
thf(fact_6556_or_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ A )
      = A ) ).

% or.idem
thf(fact_6557_or_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ A )
      = A ) ).

% or.idem
thf(fact_6558_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ X )
        = ( sqrt @ Y ) )
      = ( X = Y ) ) ).

% real_sqrt_eq_iff
thf(fact_6559_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_6560_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_6561_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_6562_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_6563_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_6564_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_6565_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_6566_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_6567_or_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
      = A ) ).

% or.right_neutral
thf(fact_6568_or_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
      = A ) ).

% or.right_neutral
thf(fact_6569_or_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
      = A ) ).

% or.left_neutral
thf(fact_6570_or_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
      = A ) ).

% or.left_neutral
thf(fact_6571_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_6572_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_6573_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_6574_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_6575_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_6576_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_6577_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel_iff
thf(fact_6578_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_less_mono
thf(fact_6579_take__bit__and,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).

% take_bit_and
thf(fact_6580_take__bit__and,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) ) ).

% take_bit_and
thf(fact_6581_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_6582_take__bit__or,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).

% take_bit_or
thf(fact_6583_take__bit__or,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) ) ).

% take_bit_or
thf(fact_6584_concat__bit__of__zero__2,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_6585_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_6586_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_6587_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_6588_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_6589_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_6590_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_6591_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6592_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6593_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_6594_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_6595_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_6596_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_6597_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_6598_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_6599_bit_Odisj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_right
thf(fact_6600_bit_Odisj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_right
thf(fact_6601_bit_Odisj__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_left
thf(fact_6602_bit_Odisj__one__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_left
thf(fact_6603_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_6604_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_gt_0_iff
thf(fact_6605_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_6606_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_6607_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_6608_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_6609_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_6610_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_6611_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_6612_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_6613_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_6614_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_6615_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_6616_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_6617_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_6618_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_6619_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_6620_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_6621_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_6622_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_6623_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_6624_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_6625_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_6626_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_6627_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_6628_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_6629_exp__less__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_6630_one__less__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_less_exp_iff
thf(fact_6631_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_6632_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_6633_take__bit__minus__one__eq__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N2 ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_6634_take__bit__minus__one__eq__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_6635_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X ) )
        = X )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% exp_ln_iff
thf(fact_6636_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_6637_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_6638_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_6639_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_6640_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_6641_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_6642_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6643_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6644_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_6645_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_6646_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6647_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6648_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6649_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6650_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6651_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_6652_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_6653_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_6654_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_6655_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_6656_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_6657_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_6658_and__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_6659_and__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_6660_or__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_6661_or__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_6662_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_6663_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_6664_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_6665_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_6666_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_6667_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_6668_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_6669_even__take__bit__eq,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_6670_even__take__bit__eq,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_6671_even__take__bit__eq,axiom,
    ! [N2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_6672_real__sqrt__abs,axiom,
    ! [X: real] :
      ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs
thf(fact_6673_and__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_6674_and__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_6675_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_6676_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_6677_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_6678_take__bit__Suc__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_6679_take__bit__Suc__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_6680_take__bit__Suc__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_6681_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_6682_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_6683_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_6684_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6685_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6686_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6687_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_6688_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6689_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_6690_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6691_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_6692_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N2 @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_6693_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_6694_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_6695_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_6696_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_6697_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_6698_take__bit__eq__mask,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_6699_take__bit__eq__mask,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_6700_take__bit__of__int,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_of_int
thf(fact_6701_of__int__and__eq,axiom,
    ! [K: int,L: int] :
      ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).

% of_int_and_eq
thf(fact_6702_of__int__or__eq,axiom,
    ! [K: int,L: int] :
      ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).

% of_int_or_eq
thf(fact_6703_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_6704_bit_Odisj__conj__distrib2,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ Z ) @ X )
      = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ X ) @ ( bit_se1409905431419307370or_int @ Z @ X ) ) ) ).

% bit.disj_conj_distrib2
thf(fact_6705_bit_Oconj__disj__distrib2,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ Z ) @ X )
      = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z @ X ) ) ) ).

% bit.conj_disj_distrib2
thf(fact_6706_bit_Odisj__conj__distrib,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y @ Z ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Z ) ) ) ).

% bit.disj_conj_distrib
thf(fact_6707_bit_Oconj__disj__distrib,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y @ Z ) )
      = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z ) ) ) ).

% bit.conj_disj_distrib
thf(fact_6708_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.left_commute
thf(fact_6709_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.left_commute
thf(fact_6710_or_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C ) )
      = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).

% or.left_commute
thf(fact_6711_or_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
      = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).

% or.left_commute
thf(fact_6712_and_Ocommute,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [A3: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A3 ) ) ) ).

% and.commute
thf(fact_6713_and_Ocommute,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [A3: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A3 ) ) ) ).

% and.commute
thf(fact_6714_or_Ocommute,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [A3: int,B2: int] : ( bit_se1409905431419307370or_int @ B2 @ A3 ) ) ) ).

% or.commute
thf(fact_6715_or_Ocommute,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [A3: nat,B2: nat] : ( bit_se1412395901928357646or_nat @ B2 @ A3 ) ) ) ).

% or.commute
thf(fact_6716_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.assoc
thf(fact_6717_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.assoc
thf(fact_6718_or_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C )
      = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).

% or.assoc
thf(fact_6719_or_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
      = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).

% or.assoc
thf(fact_6720_take__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_6721_take__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_6722_bit_Odisj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
      = X ) ).

% bit.disj_zero_right
thf(fact_6723_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se1409905431419307370or_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
thf(fact_6724_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( bit_se1412395901928357646or_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
thf(fact_6725_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_6726_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = ( bit_se2925701944663578781it_nat @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_6727_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_6728_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_6729_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_6730_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel
thf(fact_6731_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_6732_real__sqrt__divide,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_divide
thf(fact_6733_real__sqrt__mult,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_mult
thf(fact_6734_real__sqrt__power,axiom,
    ! [X: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X @ K ) )
      = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).

% real_sqrt_power
thf(fact_6735_real__sqrt__minus,axiom,
    ! [X: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_minus
thf(fact_6736_take__bit__minus,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_6737_exp__times__arg__commute,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
      = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).

% exp_times_arg_commute
thf(fact_6738_exp__times__arg__commute,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
      = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).

% exp_times_arg_commute
thf(fact_6739_take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_6740_take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_6741_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,X: code_integer,Y: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ X )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ A @ X )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
            = zero_z3403309356797280102nteger )
         => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_6742_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ X )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ A @ X )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
            = zero_zero_int )
         => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
              = ( uminus_uminus_int @ one_one_int ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_6743_concat__bit__take__bit__eq,axiom,
    ! [N2: nat,B: int] :
      ( ( bit_concat_bit @ N2 @ ( bit_se2923211474154528505it_int @ N2 @ B ) )
      = ( bit_concat_bit @ N2 @ B ) ) ).

% concat_bit_take_bit_eq
thf(fact_6744_concat__bit__eq__iff,axiom,
    ! [N2: nat,K: int,L: int,R2: int,S: int] :
      ( ( ( bit_concat_bit @ N2 @ K @ L )
        = ( bit_concat_bit @ N2 @ R2 @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = ( bit_se2923211474154528505it_int @ N2 @ R2 ) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_6745_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_6746_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_6747_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_gt_zero
thf(fact_6748_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y ) ) ).

% exp_total
thf(fact_6749_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_6750_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_6751_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_6752_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_6753_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_6754_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_6755_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_6756_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_6757_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_6758_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_6759_OR__lower,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 @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_6760_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_6761_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_6762_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_6763_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_6764_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_6765_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_6766_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ A )
        = ( bit_ri631733984087533419it_int @ N2 @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_6767_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_6768_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% take_bit_nonnegative
thf(fact_6769_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_6770_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_6771_signed__take__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_6772_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
        = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_6773_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
        = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_6774_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
      = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_6775_mult__exp__exp,axiom,
    ! [X: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_6776_exp__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).

% exp_diff
thf(fact_6777_exp__diff,axiom,
    ! [X: real,Y: real] :
      ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
      = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_diff
thf(fact_6778_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_6779_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_6780_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_6781_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_6782_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_6783_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_6784_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: complex > nat,F: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6785_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6786_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X2: set_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6787_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6788_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6789_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6790_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6791_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6792_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6793_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_6794_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6795_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6796_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6797_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N2 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_6798_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6799_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6800_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6801_sum__diff__nat,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6802_sum__diff__nat,axiom,
    ! [B3: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6803_sum__diff__nat,axiom,
    ! [B3: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6804_exp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).

% exp_gt_one
thf(fact_6805_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_6806_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_6807_take__bit__signed__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_6808_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_6809_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_6810_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_6811_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_6812_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_6813_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_6814_exp__minus__inverse,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_6815_exp__minus__inverse,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_6816_take__bit__decr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_6817_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6818_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6819_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6820_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6821_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6822_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6823_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_6824_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_6825_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6826_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6827_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6828_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_6829_even__or__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_6830_even__or__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_6831_even__or__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_6832_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_6833_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6834_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6835_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6836_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6837_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6838_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6839_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6840_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6841_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6842_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6843_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_6844_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_6845_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6846_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6847_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_complex @ ( G @ ( suc @ N2 ) ) @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6848_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6849_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6850_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6851_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6852_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_complex @ ( G @ M ) @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6853_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6854_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6855_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_6856_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_6857_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6858_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6859_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_complex @ ( G @ M )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6860_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6861_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6862_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6863_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6864_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6865_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_6866_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_6867_take__bit__eq__mod,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_6868_take__bit__eq__mod,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_6869_take__bit__eq__mod,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_6870_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6871_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6872_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6873_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6874_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6875_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6876_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_6877_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_6878_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_6879_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_6880_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_6881_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_6882_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_nat_def
thf(fact_6883_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_6884_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6885_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6886_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > complex,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6887_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6888_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6889_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_6890_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_int_def
thf(fact_6891_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_6892_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_6893_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_6894_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N2 @ A )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6895_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6896_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6897_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6898_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6899_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_6900_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_6901_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_6902_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_6903_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_6904_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_6905_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_6906_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_6907_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_6908_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_6909_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_6910_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_6911_mask__Suc__exp,axiom,
    ! [N2: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N2 ) )
      = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% mask_Suc_exp
thf(fact_6912_mask__Suc__exp,axiom,
    ! [N2: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N2 ) )
      = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).

% mask_Suc_exp
thf(fact_6913_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_6914_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_6915_exp__double,axiom,
    ! [Z: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_6916_exp__double,axiom,
    ! [Z: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_6917_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_6918_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_6919_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_6920_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_6921_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6922_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6923_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6924_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_6925_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_6926_or__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6927_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6928_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_6929_one__or__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6930_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6931_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_6932_sqrt__even__pow2,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_6933_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N2 ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_6934_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N2 ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_6935_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_6936_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_6937_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_6938_ln__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( sqrt @ X ) )
        = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_6939_OR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_6940_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_6941_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_6942_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_6943_take__bit__incr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_6944_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_6945_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6946_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6947_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6948_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6949_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6950_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6951_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6952_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6953_sum_Oin__pairs,axiom,
    ! [G: nat > complex,M: nat,N2: nat] :
      ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( plus_plus_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6954_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6955_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_6956_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_6957_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_6958_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_6959_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_6960_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_6961_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_6962_take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6963_take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6964_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6965_arsinh__real__aux,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_6966_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_6967_real__sqrt__power__even,axiom,
    ! [N2: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N2 )
          = ( power_power_real @ X @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_6968_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_6969_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_6970_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_6971_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_6972_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_6973_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_6974_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_6975_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_6976_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N2 @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N2 @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6977_stable__imp__take__bit__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6978_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6979_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_6980_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_6981_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_6982_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_6983_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_6984_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_6985_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_6986_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_6987_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_6988_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_6989_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_6990_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_6991_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_6992_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_6993_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_6994_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_6995_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_6996_or__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_6997_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_6998_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_6999_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_7000_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_7001_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_7002_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7003_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7004_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7005_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7006_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7007_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7008_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7009_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7010_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7011_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7012_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_7013_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_7014_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_7015_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_7016_set__replicate,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_7017_set__replicate,axiom,
    ! [N2: nat,X: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_7018_set__replicate,axiom,
    ! [N2: nat,X: int] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_7019_set__replicate,axiom,
    ! [N2: nat,X: real] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_7020_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7021_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7022_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7023_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7024_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7025_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7026_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7027_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7028_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7029_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7030_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7031_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( numeral_numeral_real @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7032_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7033_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7034_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7035_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7036_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7037_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7038_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7039_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7040_Suc__0__and__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_7041_and__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_7042_or__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_7043_or__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_7044_or__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_7045_or__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_7046_or__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_7047_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_7048_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_7049_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_7050_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_7051_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_7052_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_7053_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(3)
thf(fact_7054_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_7055_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_7056_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_7057_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_7058_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_7059_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_7060_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_7061_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_7062_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S4 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7063_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7064_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7065_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y4: int] :
                    ( ( member_int @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7066_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7067_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S4: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S4 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7068_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7069_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7070_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y4: int] :
                    ( ( member_int @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7071_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X3 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_7072_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B6: real,A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A6 )
                   => ( ord_less_real @ B6 @ X5 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_real @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7073_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B6: rat,A6: set_rat] :
              ( ( finite_finite_rat @ A6 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A6 )
                   => ( ord_less_rat @ B6 @ X5 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_rat @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7074_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B6: num,A6: set_num] :
              ( ( finite_finite_num @ A6 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A6 )
                   => ( ord_less_num @ B6 @ X5 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_num @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7075_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B6: nat,A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A6 )
                   => ( ord_less_nat @ B6 @ X5 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_nat @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7076_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B6: int,A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A6 )
                   => ( ord_less_int @ B6 @ X5 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_int @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7077_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B6: real,A6: set_real] :
              ( ( finite_finite_real @ A6 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A6 )
                   => ( ord_less_real @ X5 @ B6 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_real @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7078_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B6: rat,A6: set_rat] :
              ( ( finite_finite_rat @ A6 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A6 )
                   => ( ord_less_rat @ X5 @ B6 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_rat @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7079_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B6: num,A6: set_num] :
              ( ( finite_finite_num @ A6 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A6 )
                   => ( ord_less_num @ X5 @ B6 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_num @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7080_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B6: nat,A6: set_nat] :
              ( ( finite_finite_nat @ A6 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A6 )
                   => ( ord_less_nat @ X5 @ B6 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_nat @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7081_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B6: int,A6: set_int] :
              ( ( finite_finite_int @ A6 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A6 )
                   => ( ord_less_int @ X5 @ B6 ) )
               => ( ( P @ A6 )
                 => ( P @ ( insert_int @ B6 @ A6 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7082_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups2240296850493347238T_real @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7083_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7084_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7085_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7086_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups136491112297645522BT_rat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7087_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7088_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7089_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7090_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7091_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups771621172384141258BT_nat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7092_sum__diff1__nat,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ( member_VEBT_VEBT @ A @ A2 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ A @ A2 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( groups771621172384141258BT_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7093_sum__diff1__nat,axiom,
    ! [A: complex,A2: set_complex,F: complex > nat] :
      ( ( ( member_complex @ A @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_complex @ A @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7094_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7095_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7096_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7097_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7098_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_7099_set__update__subset__insert,axiom,
    ! [Xs2: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_7100_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_7101_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_7102_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_7103_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(2)
thf(fact_7104_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_7105_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_real @ ( inc @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_7106_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_7107_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_7108_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_7109_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_7110_set__replicate__Suc,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_7111_set__replicate__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_7112_set__replicate__Suc,axiom,
    ! [N2: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_7113_set__replicate__Suc,axiom,
    ! [N2: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_7114_set__replicate__conv__if,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7115_set__replicate__conv__if,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7116_set__replicate__conv__if,axiom,
    ! [N2: nat,X: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
          = bot_bot_set_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7117_set__replicate__conv__if,axiom,
    ! [N2: nat,X: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
          = bot_bot_set_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7118_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7119_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7120_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7121_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7122_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7123_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7124_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > int,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups769130701875090982BT_int @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7125_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7126_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > complex,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups1794756597179926696omplex @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_complex @ ( G @ X ) @ ( groups1794756597179926696omplex @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7127_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > real,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7128_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7129_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7130_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7131_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7132_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups771621172384141258BT_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7133_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7134_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups769130701875090982BT_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7135_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7136_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups1794756597179926696omplex @ G @ A2 )
          = ( plus_plus_complex @ ( G @ X ) @ ( groups1794756597179926696omplex @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7137_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7138_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups2240296850493347238T_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7139_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7140_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7141_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7142_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups136491112297645522BT_rat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups136491112297645522BT_rat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7143_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7144_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7145_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7146_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups769130701875090982BT_int @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_int @ ( groups769130701875090982BT_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups769130701875090982BT_int @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups769130701875090982BT_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7147_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7148_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M3: num] :
                ( ( Xa2
                  = ( bit0 @ M3 ) )
               => ( Y
                 != ( bit1 @ M3 ) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa2
                    = ( bit1 @ M3 ) )
                 => ( Y
                   != ( bit1 @ M3 ) ) ) )
           => ( ( ? [N: num] :
                    ( X
                    = ( bit0 @ N ) )
               => ( ( Xa2 = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N: num] :
                    ( ( X
                      = ( bit0 @ N ) )
                   => ! [M3: num] :
                        ( ( Xa2
                          = ( bit0 @ M3 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N @ M3 ) ) ) ) )
               => ( ! [N: num] :
                      ( ( X
                        = ( bit0 @ N ) )
                     => ! [M3: num] :
                          ( ( Xa2
                            = ( bit1 @ M3 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N @ M3 ) ) ) ) )
                 => ( ( ? [N: num] :
                          ( X
                          = ( bit1 @ N ) )
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( X
                            = ( bit1 @ N ) )
                         => ! [M3: num] :
                              ( ( Xa2
                                = ( bit0 @ M3 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N @ M3 ) ) ) ) )
                     => ~ ! [N: num] :
                            ( ( X
                              = ( bit1 @ N ) )
                           => ! [M3: num] :
                                ( ( Xa2
                                  = ( bit1 @ M3 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_7149_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real,C: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7150_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7151_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat,C: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K2: vEBT_VEBT] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K2: vEBT_VEBT] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7152_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7153_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > nat,C: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K2: vEBT_VEBT] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K2: vEBT_VEBT] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7154_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7155_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > int,C: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K2: vEBT_VEBT] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K2: vEBT_VEBT] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7156_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7157_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > complex,C: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K2: vEBT_VEBT] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_complex @ ( B @ A ) @ ( groups1794756597179926696omplex @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K2: vEBT_VEBT] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1794756597179926696omplex @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7158_sum_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7159_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups2240296850493347238T_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7160_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7161_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > real] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7162_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7163_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7164_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7165_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7166_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7167_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > rat] :
      ( ( member_nat @ I @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7168_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups771621172384141258BT_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7169_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_7170_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_7171_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_7172_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_7173_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_7174_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_7175_singleton__conv,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X2: vEBT_VEBT] : X2 = A )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv
thf(fact_7176_singleton__conv,axiom,
    ! [A: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] : X2 = A )
      = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv
thf(fact_7177_singleton__conv,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ^ [X2: complex] : X2 = A )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_7178_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] : X2 = A )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_7179_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : X2 = A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_7180_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X2: int] : X2 = A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_7181_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X2: real] : X2 = A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_7182_singleton__conv2,axiom,
    ! [A: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ( ^ [Y5: vEBT_VEBT,Z5: vEBT_VEBT] : Y5 = Z5
          @ A ) )
      = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv2
thf(fact_7183_singleton__conv2,axiom,
    ! [A: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ( ^ [Y5: product_prod_int_int,Z5: product_prod_int_int] : Y5 = Z5
          @ A ) )
      = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv2
thf(fact_7184_singleton__conv2,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ( ^ [Y5: complex,Z5: complex] : Y5 = Z5
          @ A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_7185_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y5: set_nat,Z5: set_nat] : Y5 = Z5
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_7186_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z5: nat] : Y5 = Z5
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_7187_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z5: int] : Y5 = Z5
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_7188_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y5: real,Z5: real] : Y5 = Z5
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_7189_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_7190_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_7191_sum__gp,axiom,
    ! [N2: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7192_sum__gp,axiom,
    ! [N2: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7193_sum__gp,axiom,
    ! [N2: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7194_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7195_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7196_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7197_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7198_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_7199_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_7200_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_7201_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_7202_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_7203_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7204_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_7205_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7206_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_7207_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7208_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7209_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7210_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7211_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7212_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7213_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_7214_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_7215_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_7216_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_7217_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_7218_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7219_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7220_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7221_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7222_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% of_nat_numeral
thf(fact_7223_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% of_nat_numeral
thf(fact_7224_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% of_nat_numeral
thf(fact_7225_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% of_nat_numeral
thf(fact_7226_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6690914467698888265omplex @ N2 ) ) ).

% of_nat_numeral
thf(fact_7227_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7228_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7229_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7230_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7231_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_add
thf(fact_7232_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_7233_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_add
thf(fact_7234_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_7235_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_add
thf(fact_7236_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ N2 )
        = one_one_rat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7237_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7238_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ N2 )
        = one_one_real )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7239_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7240_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7241_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7242_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7243_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7244_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7245_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7246_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_7247_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_7248_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_7249_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_7250_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_7251_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7252_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7253_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7254_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7255_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7256_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_7257_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_7258_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_7259_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_7260_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7261_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7262_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7263_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7264_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7265_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7266_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7267_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7268_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7269_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_7270_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_7271_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_7272_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_7273_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_7274_of__nat__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7275_of__nat__sum,axiom,
    ! [F: complex > nat,A2: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7276_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7277_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7278_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7279_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7280_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7281_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7282_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_7283_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_7284_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7285_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_7286_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_7287_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7288_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_7289_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_7290_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_7291_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_7292_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_7293_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_7294_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_7295_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_7296_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7297_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7298_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7299_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7300_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7301_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7302_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7303_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7304_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7305_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7306_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7307_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7308_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7309_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7310_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7311_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7312_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7313_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7314_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7315_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7316_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7317_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7318_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7319_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7320_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7321_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7322_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7323_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7324_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7325_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7326_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_7327_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_7328_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_7329_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7330_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7331_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7332_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7333_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7334_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7335_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7336_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7337_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7338_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7339_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7340_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7341_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7342_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7343_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7344_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7345_set__encode__insert,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N2 @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_7346_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% real_arch_simple
thf(fact_7347_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% real_arch_simple
thf(fact_7348_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% reals_Archimedean2
thf(fact_7349_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% reals_Archimedean2
thf(fact_7350_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7351_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_7352_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7353_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_7354_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7355_of__nat__and__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N2 ) )
      = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_and_eq
thf(fact_7356_of__nat__and__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N2 ) )
      = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_and_eq
thf(fact_7357_of__nat__or__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se1412395901928357646or_nat @ M @ N2 ) )
      = ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_or_eq
thf(fact_7358_of__nat__or__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se1412395901928357646or_nat @ M @ N2 ) )
      = ( bit_se1412395901928357646or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_or_eq
thf(fact_7359_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_7360_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_7361_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_7362_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_7363_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_7364_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_7365_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_7366_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_7367_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_7368_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_7369_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_7370_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_7371_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_7372_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_7373_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_7374_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_7375_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_7376_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_7377_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7378_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7379_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7380_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7381_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7382_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7383_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7384_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7385_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_7386_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_7387_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_7388_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_7389_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_7390_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_7391_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_7392_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_7393_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_7394_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_7395_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% int_cases
thf(fact_7396_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N: nat] : ( P @ ( semiri1314217659103216013at_int @ N ) )
     => ( ! [N: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_7397_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_7398_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_7399_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_7400_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_mod
thf(fact_7401_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mod
thf(fact_7402_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mod
thf(fact_7403_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_7404_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_7405_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_7406_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_7407_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_7408_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_7409_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y ) ) ) ).

% of_nat_max
thf(fact_7410_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_7411_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_7412_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_7413_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_7414_take__bit__of__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N2 @ M ) ) ) ).

% take_bit_of_nat
thf(fact_7415_take__bit__of__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) ) ) ).

% take_bit_of_nat
thf(fact_7416_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_7417_of__nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% of_nat_mask_eq
thf(fact_7418_of__nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% of_nat_mask_eq
thf(fact_7419_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_7420_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7421_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7422_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7423_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7424_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7425_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7426_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7427_exp__of__nat2__mult,axiom,
    ! [X: real,N2: nat] :
      ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).

% exp_of_nat2_mult
thf(fact_7428_exp__of__nat2__mult,axiom,
    ! [X: complex,N2: nat] :
      ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) )
      = ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).

% exp_of_nat2_mult
thf(fact_7429_exp__of__nat__mult,axiom,
    ! [N2: nat,X: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) )
      = ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).

% exp_of_nat_mult
thf(fact_7430_exp__of__nat__mult,axiom,
    ! [N2: nat,X: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ X ) )
      = ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).

% exp_of_nat_mult
thf(fact_7431_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y4: real] :
        ? [N: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_7432_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% int_cases4
thf(fact_7433_real__of__nat__div4,axiom,
    ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_7434_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_7435_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_7436_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_7437_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_7438_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M @ N2 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_7439_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_7440_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_7441_real__of__nat__div,axiom,
    ! [D: nat,N2: nat] :
      ( ( dvd_dvd_nat @ D @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_7442_int__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_7443_int__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_7444_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7445_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7446_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7447_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_7448_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_7449_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_7450_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% pos_int_cases
thf(fact_7451_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N )
          & ( K
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_7452_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) )
       => ~ ! [N: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% int_cases3
thf(fact_7453_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).

% nat_less_real_le
thf(fact_7454_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N3: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_7455_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_7456_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_7457_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% negD
thf(fact_7458_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7459_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_7460_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_7461_empty__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X2: product_prod_int_int] : $false ) ) ).

% empty_def
thf(fact_7462_empty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X2: complex] : $false ) ) ).

% empty_def
thf(fact_7463_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X2: set_nat] : $false ) ) ).

% empty_def
thf(fact_7464_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $false ) ) ).

% empty_def
thf(fact_7465_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X2: int] : $false ) ) ).

% empty_def
thf(fact_7466_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X2: real] : $false ) ) ).

% empty_def
thf(fact_7467_insert__Collect,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( insert_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( collect_VEBT_VEBT
        @ ^ [U2: vEBT_VEBT] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7468_insert__Collect,axiom,
    ! [A: real,P: real > $o] :
      ( ( insert_real @ A @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7469_insert__Collect,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( insert5033312907999012233nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7470_insert__Collect,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( insert_complex @ A @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7471_insert__Collect,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7472_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7473_insert__Collect,axiom,
    ! [A: int,P: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7474_insert__compr,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A3: vEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X2: vEBT_VEBT] :
              ( ( X2 = A3 )
              | ( member_VEBT_VEBT @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7475_insert__compr,axiom,
    ( insert_real
    = ( ^ [A3: real,B5: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ( ( X2 = A3 )
              | ( member_real @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7476_insert__compr,axiom,
    ( insert5033312907999012233nt_int
    = ( ^ [A3: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( X2 = A3 )
              | ( member5262025264175285858nt_int @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7477_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A3: complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ( ( X2 = A3 )
              | ( member_complex @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7478_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A3: set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X2: set_nat] :
              ( ( X2 = A3 )
              | ( member_set_nat @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7479_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( X2 = A3 )
              | ( member_nat @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7480_insert__compr,axiom,
    ( insert_int
    = ( ^ [A3: int,B5: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ( ( X2 = A3 )
              | ( member_int @ X2 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_7481_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7482_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7483_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_7484_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_7485_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_7486_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_7487_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_7488_exp__divide__power__eq,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
        = ( exp_real @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7489_exp__divide__power__eq,axiom,
    ! [N2: nat,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
        = ( exp_complex @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7490_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M3 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_7491_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% neg_int_cases
thf(fact_7492_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_7493_real__of__nat__div2,axiom,
    ! [N2: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_7494_real__of__nat__div3,axiom,
    ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_7495_ln__realpow,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_7496_linear__plus__1__le__power,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).

% linear_plus_1_le_power
thf(fact_7497_Bernoulli__inequality,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).

% Bernoulli_inequality
thf(fact_7498_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_7499_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K3: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
           => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K3 @ L4 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_7500_Collect__conv__if,axiom,
    ! [P: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
thf(fact_7501_Collect__conv__if,axiom,
    ! [P: product_prod_int_int > $o,A: product_prod_int_int] :
      ( ( ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if
thf(fact_7502_Collect__conv__if,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_7503_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_7504_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_7505_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_7506_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( X2 = A )
                & ( P @ X2 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_7507_Collect__conv__if2,axiom,
    ! [P: vEBT_VEBT > $o,A: vEBT_VEBT] :
      ( ( ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
thf(fact_7508_Collect__conv__if2,axiom,
    ! [P: product_prod_int_int > $o,A: product_prod_int_int] :
      ( ( ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if2
thf(fact_7509_Collect__conv__if2,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_7510_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7511_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7512_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_7513_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( A = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_7514_double__arith__series,axiom,
    ! [A: rat,D: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7515_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7516_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7517_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7518_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7519_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_7520_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_7521_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_7522_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_7523_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_7524_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_7525_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_7526_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_7527_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7528_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7529_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7530_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7531_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7532_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7533_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7534_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7535_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7536_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7537_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7538_sum__gp__offset,axiom,
    ! [X: real,M: nat,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7539_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7540_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7541_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M5: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M5 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M5 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7542_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M5: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M5 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M5 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7543_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M5: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7544_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M5: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M5 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M5 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7545_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M5: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M5 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M5 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7546_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
     => ( ( vEBT_invar_vebt @ T @ Deg )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_7547_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7548_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7549_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7550_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7551_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_7552_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_7553_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_7554_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_real @ A @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_7555_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_7556_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_7557_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_7558_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_7559_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_7560_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_7561_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_7562_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_7563_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7564_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7565_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B ) )
          = ( semiri5074537144036343181t_real @ B ) ) ) ) ).

% log_pow_cancel
thf(fact_7566_log__def,axiom,
    ( log
    = ( ^ [A3: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A3 ) ) ) ) ).

% log_def
thf(fact_7567_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_7568_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_7569_less__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_7570_log__of__power__eq,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N2 )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_7571_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_7572_norm__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).

% norm_exp
thf(fact_7573_norm__exp,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).

% norm_exp
thf(fact_7574_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X )
          = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_7575_le__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_7576_log__base__pow,axiom,
    ! [A: real,N2: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N2 ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_7577_log__nat__power,axiom,
    ! [X: real,B: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_7578_log2__of__power__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( semiri5074537144036343181t_real @ N2 )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_7579_log__of__power__less,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_7580_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_7581_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_7582_log__of__power__le,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_7583_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_7584_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_7585_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_7586_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_7587_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7588_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7589_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_7590_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_7591_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7592_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7593_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_7594_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N2: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_7595_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_7596_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7597_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_7598_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_7599_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_7600_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_7601_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_7602_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_7603_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7604_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7605_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_7606_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_7607_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_7608_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_7609_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7610_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7611_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_7612_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_7613_norm__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_7614_norm__minus__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_7615_norm__not__less__zero,axiom,
    ! [X: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_7616_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_7617_norm__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult
thf(fact_7618_norm__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult
thf(fact_7619_norm__divide,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_divide
thf(fact_7620_norm__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_divide
thf(fact_7621_sum__norm__le,axiom,
    ! [S3: set_real,F: real > complex,G: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S3 ) ) @ ( groups8097168146408367636l_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_7622_sum__norm__le,axiom,
    ! [S3: set_set_nat,F: set_nat > complex,G: set_nat > real] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S3 ) ) @ ( groups5107569545109728110t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_7623_sum__norm__le,axiom,
    ! [S3: set_int,F: int > complex,G: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S3 ) ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_7624_sum__norm__le,axiom,
    ! [S3: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_7625_sum__norm__le,axiom,
    ! [S3: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S3 ) ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_7626_sum__norm__le,axiom,
    ! [S3: set_nat,F: nat > real,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_7627_norm__power,axiom,
    ! [X: real,N2: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).

% norm_power
thf(fact_7628_norm__power,axiom,
    ! [X: complex,N2: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).

% norm_power
thf(fact_7629_norm__sum,axiom,
    ! [F: nat > complex,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_7630_norm__sum,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_7631_norm__sum,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_7632_norm__uminus__minus,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_7633_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_7634_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7635_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7636_power__eq__imp__eq__norm,axiom,
    ! [W: real,N2: nat,Z: real] :
      ( ( ( power_power_real @ W @ N2 )
        = ( power_power_real @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7637_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N2: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N2 )
        = ( power_power_complex @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7638_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_7639_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_7640_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_7641_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_7642_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7643_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7644_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_7645_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_7646_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7647_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7648_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_7649_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_7650_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7651_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7652_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7653_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7654_norm__power__ineq,axiom,
    ! [X: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_7655_norm__power__ineq,axiom,
    ! [X: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_7656_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7657_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7658_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_7659_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_7660_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7661_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7662_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_7663_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_7664_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_7665_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_7666_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_7667_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_7668_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_7669_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_7670_power__eq__1__iff,axiom,
    ! [W: real,N2: nat] :
      ( ( ( power_power_real @ W @ N2 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_7671_power__eq__1__iff,axiom,
    ! [W: complex,N2: nat] :
      ( ( ( power_power_complex @ W @ N2 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_7672_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7673_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7674_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_7675_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_7676_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_7677_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_7678_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_7679_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_7680_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7681_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7682_suminf__zero,axiom,
    ( ( suminf_complex
      @ ^ [N3: nat] : zero_zero_complex )
    = zero_zero_complex ) ).

% suminf_zero
thf(fact_7683_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N3: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7684_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N3: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7685_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N3: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7686_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_7687_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_complex ) )
       => ( ( suminf_complex @ F )
          = ( groups2073611262835488442omplex @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_7688_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_7689_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_7690_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_7691_log__ceil__idem,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_7692_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_rat @ N3 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_7693_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_real @ N3 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_7694_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_7695_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_7696_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_7697_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_7698_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_7699_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_7700_ceiling__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_7701_ceiling__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_7702_ceiling__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_7703_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_7704_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% zero_less_ceiling
thf(fact_7705_zero__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_ceiling
thf(fact_7706_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_7707_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_7708_ceiling__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_7709_ceiling__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_7710_one__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% one_le_ceiling
thf(fact_7711_one__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_le_ceiling
thf(fact_7712_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_7713_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_7714_ceiling__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_7715_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_7716_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_7717_one__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ one_one_real @ X ) ) ).

% one_less_ceiling
thf(fact_7718_ceiling__add__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_7719_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_7720_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7721_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7722_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_7723_ceiling__add__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_7724_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_7725_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_7726_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_7727_ceiling__diff__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_7728_ceiling__numeral__power,axiom,
    ! [X: num,N2: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).

% ceiling_numeral_power
thf(fact_7729_ceiling__numeral__power,axiom,
    ! [X: num,N2: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).

% ceiling_numeral_power
thf(fact_7730_ceiling__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_7731_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_7732_zero__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).

% zero_le_ceiling
thf(fact_7733_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_7734_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_7735_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_7736_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_7737_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_7738_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_7739_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7740_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7741_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_7742_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_7743_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_7744_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7745_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7746_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_7747_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_7748_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).

% ceiling_mono
thf(fact_7749_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_7750_le__of__int__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_7751_le__of__int__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_7752_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_7753_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_7754_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_7755_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_7756_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).

% ceiling_le
thf(fact_7757_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).

% ceiling_le
thf(fact_7758_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_7759_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_7760_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_7761_ceiling__add__le,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_7762_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_7763_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_7764_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_7765_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_7766_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7767_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7768_ceiling__correct,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
      & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% ceiling_correct
thf(fact_7769_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_7770_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_7771_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_7772_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_7773_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_7774_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_7775_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_7776_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_7777_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_7778_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_7779_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_7780_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_7781_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% le_ceiling_iff
thf(fact_7782_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_7783_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_7784_ceiling__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_7785_ceiling__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_7786_ceiling__eq,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_7787_ceiling__eq,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_7788_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_7789_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_7790_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_7791_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_7792_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_7793_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N2: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N2 ) @ ( power_power_rat @ Z @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7794_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7795_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7796_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7797_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_7798_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7799_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7800_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_7801_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_7802_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7803_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_7804_lessThan__iff,axiom,
    ! [I: $o,K: $o] :
      ( ( member_o @ I @ ( set_ord_lessThan_o @ K ) )
      = ( ord_less_o @ I @ K ) ) ).

% lessThan_iff
thf(fact_7805_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_7806_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_7807_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7808_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7809_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7810_summable__zero,axiom,
    ( summable_complex
    @ ^ [N3: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_7811_summable__zero,axiom,
    ( summable_real
    @ ^ [N3: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7812_summable__zero,axiom,
    ( summable_nat
    @ ^ [N3: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7813_summable__zero,axiom,
    ( summable_int
    @ ^ [N3: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7814_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7815_summable__iff__shift,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( summable_complex @ F ) ) ).

% summable_iff_shift
thf(fact_7816_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_7817_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_7818_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_7819_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_7820_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_7821_lessThan__subset__iff,axiom,
    ! [X: $o,Y: $o] :
      ( ( ord_less_eq_set_o @ ( set_ord_lessThan_o @ X ) @ ( set_ord_lessThan_o @ Y ) )
      = ( ord_less_eq_o @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_7822_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_7823_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
      = ( X != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_7824_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_7825_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_7826_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7827_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7828_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7829_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7830_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_7831_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_7832_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_7833_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_7834_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_7835_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_7836_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_7837_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_7838_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7839_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7840_sum_OlessThan__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups2073611262835488442omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7841_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7842_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7843_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_7844_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_7845_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_7846_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_7847_numeral__powr__numeral__real,axiom,
    ! [M: num,N2: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_powr_numeral_real
thf(fact_7848_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A @ ( log @ A @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_7849_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_7850_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_7851_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_7852_powr__numeral,axiom,
    ! [X: real,N2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N2 ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% powr_numeral
thf(fact_7853_square__powr__half,axiom,
    ! [X: real] :
      ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% square_powr_half
thf(fact_7854_summable__norm__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) )
     => ( summable_real @ F ) ) ).

% summable_norm_cancel
thf(fact_7855_summable__norm__cancel,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) )
     => ( summable_complex @ F ) ) ).

% summable_norm_cancel
thf(fact_7856_powr__powr,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X @ A ) @ B )
      = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_7857_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N7 @ N )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7858_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N7 @ N )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7859_summable__comparison__test_H,axiom,
    ! [G: nat > real,N5: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N5 @ N )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N ) ) @ ( G @ N ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7860_summable__comparison__test_H,axiom,
    ! [G: nat > real,N5: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N5 @ N )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7861_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_7862_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7863_summable__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) ) ) ).

% summable_mult
thf(fact_7864_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) ) ) ).

% summable_mult
thf(fact_7865_summable__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C ) ) ) ).

% summable_mult2
thf(fact_7866_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ).

% summable_mult2
thf(fact_7867_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7868_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7869_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7870_summable__add,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N3: nat] : ( plus_plus_complex @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7871_summable__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_diff
thf(fact_7872_summable__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_diff
thf(fact_7873_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) ) ) ).

% summable_divide
thf(fact_7874_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) ) ) ).

% summable_divide
thf(fact_7875_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7876_summable__Suc__iff,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
      = ( summable_complex @ F ) ) ).

% summable_Suc_iff
thf(fact_7877_summable__minus__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( uminus_uminus_real @ ( F @ N3 ) ) )
      = ( summable_real @ F ) ) ).

% summable_minus_iff
thf(fact_7878_summable__minus__iff,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( F @ N3 ) ) )
      = ( summable_complex @ F ) ) ).

% summable_minus_iff
thf(fact_7879_summable__minus,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( uminus_uminus_real @ ( F @ N3 ) ) ) ) ).

% summable_minus
thf(fact_7880_summable__minus,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( F @ N3 ) ) ) ) ).

% summable_minus
thf(fact_7881_summable__sum,axiom,
    ! [I5: set_complex,F: complex > nat > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N3: nat] :
            ( groups5808333547571424918x_real
            @ ^ [I3: complex] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7882_summable__sum,axiom,
    ! [I5: set_real,F: real > nat > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N3: nat] :
            ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7883_summable__sum,axiom,
    ! [I5: set_int,F: int > nat > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N3: nat] :
            ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7884_summable__sum,axiom,
    ! [I5: set_real,F: real > nat > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] :
            ( groups5754745047067104278omplex
            @ ^ [I3: real] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7885_summable__sum,axiom,
    ! [I5: set_nat,F: nat > nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] :
            ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7886_summable__sum,axiom,
    ! [I5: set_int,F: int > nat > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] :
            ( groups3049146728041665814omplex
            @ ^ [I3: int] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7887_summable__sum,axiom,
    ! [I5: set_int,F: int > nat > int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_int @ ( F @ I2 ) ) )
     => ( summable_int
        @ ^ [N3: nat] :
            ( groups4538972089207619220nt_int
            @ ^ [I3: int] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7888_summable__sum,axiom,
    ! [I5: set_complex,F: complex > nat > complex] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] :
            ( groups7754918857620584856omplex
            @ ^ [I3: complex] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7889_summable__sum,axiom,
    ! [I5: set_nat,F: nat > nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_nat @ ( F @ I2 ) ) )
     => ( summable_nat
        @ ^ [N3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7890_summable__sum,axiom,
    ! [I5: set_nat,F: nat > nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( F @ I3 @ N3 )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_7891_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7892_summable__ignore__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7893_suminf__le__const,axiom,
    ! [F: nat > int,X: int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7894_suminf__le__const,axiom,
    ! [F: nat > nat,X: nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7895_suminf__le__const,axiom,
    ! [F: nat > real,X: real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7896_summable__rabs__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
     => ( summable_real @ F ) ) ).

% summable_rabs_cancel
thf(fact_7897_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X2: set_nat] : ( ord_less_set_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7898_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7899_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X2: num] : ( ord_less_num @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7900_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7901_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7902_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7903_lessThan__def,axiom,
    ( set_ord_lessThan_o
    = ( ^ [U2: $o] :
          ( collect_o
          @ ^ [X2: $o] : ( ord_less_o @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7904_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X: int] :
      ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7905_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7906_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ X )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7907_powser__insidea,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7908_powser__insidea,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7909_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7910_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7911_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7912_powr__non__neg,axiom,
    ! [A: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_7913_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_7914_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_7915_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_7916_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_7917_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_less_mono
thf(fact_7918_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_mono
thf(fact_7919_suminf__split__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex @ F )
        = ( plus_plus_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
          @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7920_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7921_suminf__minus__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7922_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7923_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
      = ( ord_less_rat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7924_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7925_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7926_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7927_lessThan__strict__subset__iff,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
      = ( ord_less_real @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7928_lessThan__strict__subset__iff,axiom,
    ! [M: $o,N2: $o] :
      ( ( ord_less_set_o @ ( set_ord_lessThan_o @ M ) @ ( set_ord_lessThan_o @ N2 ) )
      = ( ord_less_o @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7929_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7930_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7931_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7932_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7933_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7934_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7935_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7936_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7937_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_7938_sum__less__suminf,axiom,
    ! [F: nat > int,N2: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7939_sum__less__suminf,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7940_sum__less__suminf,axiom,
    ! [F: nat > real,N2: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7941_suminf__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
        = ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).

% suminf_mult
thf(fact_7942_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_7943_suminf__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
        = ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7944_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7945_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7946_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7947_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7948_suminf__add,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( ( plus_plus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
          = ( suminf_complex
            @ ^ [N3: nat] : ( plus_plus_complex @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7949_suminf__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
          = ( suminf_complex
            @ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7950_suminf__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7951_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7952_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7953_suminf__minus,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( uminus_uminus_real @ ( F @ N3 ) ) )
        = ( uminus_uminus_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_minus
thf(fact_7954_suminf__minus,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( F @ N3 ) ) )
        = ( uminus1482373934393186551omplex @ ( suminf_complex @ F ) ) ) ) ).

% suminf_minus
thf(fact_7955_suminf__sum,axiom,
    ! [I5: set_complex,F: complex > nat > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] :
              ( groups5808333547571424918x_real
              @ ^ [I3: complex] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups5808333547571424918x_real
          @ ^ [I3: complex] : ( suminf_real @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7956_suminf__sum,axiom,
    ! [I5: set_real,F: real > nat > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] :
              ( groups8097168146408367636l_real
              @ ^ [I3: real] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups8097168146408367636l_real
          @ ^ [I3: real] : ( suminf_real @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7957_suminf__sum,axiom,
    ! [I5: set_int,F: int > nat > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] :
              ( groups8778361861064173332t_real
              @ ^ [I3: int] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups8778361861064173332t_real
          @ ^ [I3: int] : ( suminf_real @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7958_suminf__sum,axiom,
    ! [I5: set_real,F: real > nat > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] :
              ( groups5754745047067104278omplex
              @ ^ [I3: real] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups5754745047067104278omplex
          @ ^ [I3: real] : ( suminf_complex @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7959_suminf__sum,axiom,
    ! [I5: set_nat,F: nat > nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( suminf_complex @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7960_suminf__sum,axiom,
    ! [I5: set_int,F: int > nat > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] :
              ( groups3049146728041665814omplex
              @ ^ [I3: int] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups3049146728041665814omplex
          @ ^ [I3: int] : ( suminf_complex @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7961_suminf__sum,axiom,
    ! [I5: set_int,F: int > nat > int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_int @ ( F @ I2 ) ) )
     => ( ( suminf_int
          @ ^ [N3: nat] :
              ( groups4538972089207619220nt_int
              @ ^ [I3: int] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups4538972089207619220nt_int
          @ ^ [I3: int] : ( suminf_int @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7962_suminf__sum,axiom,
    ! [I5: set_complex,F: complex > nat > complex] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] :
              ( groups7754918857620584856omplex
              @ ^ [I3: complex] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups7754918857620584856omplex
          @ ^ [I3: complex] : ( suminf_complex @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7963_suminf__sum,axiom,
    ! [I5: set_nat,F: nat > nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_nat @ ( F @ I2 ) ) )
     => ( ( suminf_nat
          @ ^ [N3: nat] :
              ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( suminf_nat @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7964_suminf__sum,axiom,
    ! [I5: set_nat,F: nat > nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( F @ I3 @ N3 )
              @ I5 ) )
        = ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( suminf_real @ ( F @ I3 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_7965_sum__less__suminf2,axiom,
    ! [F: nat > int,N2: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7966_sum__less__suminf2,axiom,
    ! [F: nat > nat,N2: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7967_sum__less__suminf2,axiom,
    ! [F: nat > real,N2: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7968_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7969_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7970_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7971_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7972_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7973_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7974_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7975_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7976_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7977_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_7978_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_7979_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X )
            = ( powr_real @ A @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_7980_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_7981_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_7982_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_7983_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_7984_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_7985_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_7986_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_7987_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).

% log_base_powr
thf(fact_7988_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).

% log_powr
thf(fact_7989_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_0_powser
thf(fact_7990_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_0_powser
thf(fact_7991_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_7992_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_7993_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_7994_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_7995_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7996_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7997_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7998_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7999_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_complex @ Z @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_8000_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N3 @ M ) ) @ ( power_power_real @ Z @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_8001_powr__add,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).

% powr_add
thf(fact_8002_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).

% powr_diff
thf(fact_8003_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N7 @ N )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_8004_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N7 @ N )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_8005_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_8006_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8007_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8008_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).

% summable_rabs
thf(fact_8009_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N2: real] :
      ( ! [X3: real] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
          @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8010_sum__diff__distrib,axiom,
    ! [Q: $o > nat,P: $o > nat,N2: $o] :
      ( ! [X3: $o] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ P @ ( set_ord_lessThan_o @ N2 ) ) @ ( groups8507830703676809646_o_nat @ Q @ ( set_ord_lessThan_o @ N2 ) ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X2: $o] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
          @ ( set_ord_lessThan_o @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8011_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_8012_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8013_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8014_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8015_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_8016_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_8017_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_8018_powr__realpow,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
        = ( power_power_real @ X @ N2 ) ) ) ).

% powr_realpow
thf(fact_8019_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_8020_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_8021_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_8022_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_8023_powser__inside,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_8024_powser__inside,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_8025_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_8026_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_8027_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8028_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8029_suminf__split__head,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_8030_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_8031_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8032_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8033_pi__half__neq__two,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_neq_two
thf(fact_8034_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8035_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8036_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8037_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups2073611262835488442omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_complex @ ( G @ zero_zero_nat )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8038_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8039_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8040_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8041_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8042_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8043_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8044_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8045_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8046_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N2: nat,R2: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ R2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8047_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N2: nat,R2: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ R2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8048_sumr__diff__mult__const2,axiom,
    ! [F: nat > complex,N2: nat,R2: complex] :
      ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ R2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( minus_minus_complex @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8049_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N2: nat,R2: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ R2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8050_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) ) ).

% summable_norm
thf(fact_8051_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).

% summable_norm
thf(fact_8052_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8053_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8054_powr__minus__divide,axiom,
    ! [X: real,A: real] :
      ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).

% powr_minus_divide
thf(fact_8055_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N: nat] :
              ( ( member_nat @ N @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8056_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N: nat] :
              ( ( member_nat @ N @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8057_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N: nat] :
              ( ( member_nat @ N @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8058_powr__neg__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X ) ) ) ).

% powr_neg_one
thf(fact_8059_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_8060_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_8061_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_8062_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_8063_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_8064_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8065_pi__half__less__two,axiom,
    ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_less_two
thf(fact_8066_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_8067_power__diff__1__eq,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8068_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8069_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8070_power__diff__1__eq,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_8071_one__diff__power__eq,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8072_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8073_one__diff__power__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8074_one__diff__power__eq,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_8075_geometric__sum,axiom,
    ! [X: complex,N2: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_8076_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_8077_geometric__sum,axiom,
    ! [X: real,N2: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_8078_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_8079_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_8080_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8081_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8082_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_8083_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_8084_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_8085_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_8086_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M4: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M4 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M4 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8087_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M4: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M4 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M4 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8088_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N8: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ N8 @ N9 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_8089_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N8: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ N8 @ N9 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_8090_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_8091_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8092_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R0 @ N ) ) @ M7 )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_8093_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8094_sum__gp__strict,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ N2 ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8095_sum__gp__strict,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ N2 ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8096_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8097_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8098_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8099_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8100_pi__half__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_gt_zero
thf(fact_8101_power__diff__sumr2,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8102_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8103_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8104_power__diff__sumr2,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_8105_diff__power__eq__sum,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8106_diff__power__eq__sum,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8107_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8108_diff__power__eq__sum,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8109_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_8110_m2pi__less__pi,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).

% m2pi_less_pi
thf(fact_8111_powr__def,axiom,
    ( powr_real
    = ( ^ [X2: real,A3: real] : ( if_real @ ( X2 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X2 ) ) ) ) ) ) ).

% powr_def
thf(fact_8112_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N5 @ N )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_8113_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N5 @ N )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_8114_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8115_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_8116_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8117_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > int,K5: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8118_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8119_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > real,K5: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8120_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8121_one__diff__power__eq_H,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8122_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8123_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8124_one__diff__power__eq_H,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8125_minus__pi__half__less__zero,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).

% minus_pi_half_less_zero
thf(fact_8126_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_8127_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_8128_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_8129_powr__neg__numeral,axiom,
    ! [X: real,N2: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8130_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_8131_machin__Euler,axiom,
    ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% machin_Euler
thf(fact_8132_machin,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% machin
thf(fact_8133_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8134_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8135_sin__cos__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% sin_cos_npi
thf(fact_8136_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8137_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_8138_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_8139_summable__complex__of__real,axiom,
    ! [F: nat > real] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( F @ N3 ) ) )
      = ( summable_real @ F ) ) ).

% summable_complex_of_real
thf(fact_8140_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_8141_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_8142_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_8143_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_8144_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_8145_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = one_one_complex )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_8146_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_8147_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_8148_of__real__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_mult
thf(fact_8149_of__real__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_mult
thf(fact_8150_of__real__divide,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_divide
thf(fact_8151_of__real__divide,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_divide
thf(fact_8152_of__real__add,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_add
thf(fact_8153_of__real__add,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_add
thf(fact_8154_of__real__power,axiom,
    ! [X: real,N2: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N2 ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N2 ) ) ).

% of_real_power
thf(fact_8155_of__real__power,axiom,
    ! [X: real,N2: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N2 ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N2 ) ) ).

% of_real_power
thf(fact_8156_of__real__diff,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_diff
thf(fact_8157_of__real__diff,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_diff
thf(fact_8158_sin__pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
      = ( sin_real @ X ) ) ).

% sin_pi_minus
thf(fact_8159_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8160_of__real__sum,axiom,
    ! [F: complex > real,S: set_complex] :
      ( ( real_V4546457046886955230omplex @ ( groups5808333547571424918x_real @ F @ S ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( real_V4546457046886955230omplex @ ( F @ X2 ) )
        @ S ) ) ).

% of_real_sum
thf(fact_8161_of__real__sum,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( real_V4546457046886955230omplex @ ( groups6591440286371151544t_real @ F @ S ) )
      = ( groups2073611262835488442omplex
        @ ^ [X2: nat] : ( real_V4546457046886955230omplex @ ( F @ X2 ) )
        @ S ) ) ).

% of_real_sum
thf(fact_8162_of__real__sum,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( real_V1803761363581548252l_real @ ( groups6591440286371151544t_real @ F @ S ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( real_V1803761363581548252l_real @ ( F @ X2 ) )
        @ S ) ) ).

% of_real_sum
thf(fact_8163_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_8164_cos__periodic__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi
thf(fact_8165_cos__periodic__pi2,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi2
thf(fact_8166_sin__periodic__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi
thf(fact_8167_sin__periodic__pi2,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi2
thf(fact_8168_cos__pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_pi_minus
thf(fact_8169_cos__minus__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_minus_pi
thf(fact_8170_sin__minus__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_minus_pi
thf(fact_8171_sin__cos__squared__add3,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_8172_sin__cos__squared__add3,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_8173_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_8174_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_8175_cos__of__real__pi,axiom,
    ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_of_real_pi
thf(fact_8176_cos__of__real__pi,axiom,
    ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cos_of_real_pi
thf(fact_8177_sin__npi2,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8178_sin__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8179_sin__npi__int,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8180_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_8181_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8182_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8183_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8184_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8185_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8186_norm__of__real__addn,axiom,
    ! [X: real,B: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_8187_norm__of__real__addn,axiom,
    ! [X: real,B: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_8188_cos__periodic,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X ) ) ).

% cos_periodic
thf(fact_8189_sin__periodic,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X ) ) ).

% sin_periodic
thf(fact_8190_cos__2pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( cos_real @ X ) ) ).

% cos_2pi_minus
thf(fact_8191_cos__npi2,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% cos_npi2
thf(fact_8192_cos__npi,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% cos_npi
thf(fact_8193_sin__cos__squared__add,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_8194_sin__cos__squared__add,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_8195_sin__cos__squared__add2,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_8196_sin__cos__squared__add2,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_8197_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_8198_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_8199_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_8200_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_8201_sin__2npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_8202_cos__2npi,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_8203_sin__2pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_2pi_minus
thf(fact_8204_sin__int__2pin,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8205_cos__int__2pin,axiom,
    ! [N2: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_8206_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_8207_sin__3over2__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sin_3over2_pi
thf(fact_8208_cos__npi__int,axiom,
    ! [N2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_8209_sin__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( sin_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% sin_add
thf(fact_8210_sin__add,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_8211_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A7: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X2: real] : ( member_real @ X2 @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8212_uminus__set__def,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A7: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( uminus7117520113953359693_int_o
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8213_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A7: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X2: complex] : ( member_complex @ X2 @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8214_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A7: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8215_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A7: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X2: nat] : ( member_nat @ X2 @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8216_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A7: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X2: int] : ( member_int @ X2 @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8217_Collect__neg__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] :
            ~ ( P @ X2 ) )
      = ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8218_Collect__neg__eq,axiom,
    ! [P: complex > $o] :
      ( ( collect_complex
        @ ^ [X2: complex] :
            ~ ( P @ X2 ) )
      = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8219_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] :
            ~ ( P @ X2 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8220_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ~ ( P @ X2 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8221_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X2: int] :
            ~ ( P @ X2 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8222_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A7: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ~ ( member_real @ X2 @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_8223_Compl__eq,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A7: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ~ ( member5262025264175285858nt_int @ X2 @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_8224_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A7: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ~ ( member_complex @ X2 @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_8225_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A7: set_set_nat] :
          ( collect_set_nat
          @ ^ [X2: set_nat] :
              ~ ( member_set_nat @ X2 @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_8226_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A7: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ~ ( member_nat @ X2 @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_8227_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A7: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ~ ( member_int @ X2 @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_8228_sin__diff,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_diff
thf(fact_8229_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R3: real,A5: real] :
      ( ( X
        = ( times_times_real @ R3 @ ( cos_real @ A5 ) ) )
      & ( Y
        = ( times_times_real @ R3 @ ( sin_real @ A5 ) ) ) ) ).

% polar_Ex
thf(fact_8230_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
        = one_one_complex )
     => ( ( sin_complex @ X )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_8231_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
     => ( ( sin_real @ X )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_8232_cos__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
      = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% cos_int_times_real
thf(fact_8233_cos__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
      = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% cos_int_times_real
thf(fact_8234_sin__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
      = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% sin_int_times_real
thf(fact_8235_sin__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
      = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% sin_int_times_real
thf(fact_8236_cos__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_diff
thf(fact_8237_cos__diff,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_8238_cos__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_add
thf(fact_8239_cos__add,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_8240_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8241_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( ( sin_complex @ X )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8242_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_8243_sin__cos__eq,axiom,
    ( sin_real
    = ( ^ [X2: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% sin_cos_eq
thf(fact_8244_sin__cos__eq,axiom,
    ( sin_complex
    = ( ^ [X2: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% sin_cos_eq
thf(fact_8245_cos__sin__eq,axiom,
    ( cos_real
    = ( ^ [X2: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% cos_sin_eq
thf(fact_8246_cos__sin__eq,axiom,
    ( cos_complex
    = ( ^ [X2: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% cos_sin_eq
thf(fact_8247_sin__double,axiom,
    ! [X: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).

% sin_double
thf(fact_8248_sin__double,axiom,
    ! [X: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).

% sin_double
thf(fact_8249_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8250_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_8251_minus__sin__cos__eq,axiom,
    ! [X: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8252_minus__sin__cos__eq,axiom,
    ! [X: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
      = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8253_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_8254_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8255_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).

% abs_sin_x_le_abs_x
thf(fact_8256_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_8257_summable__of__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( summable_real
        @ ^ [N3: nat] : ( real_V1803761363581548252l_real @ ( X8 @ N3 ) ) ) ) ).

% summable_of_real
thf(fact_8258_summable__of__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( summable_complex
        @ ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( X8 @ N3 ) ) ) ) ).

% summable_of_real
thf(fact_8259_cos__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_8260_cos__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_8261_sin__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_8262_sin__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_8263_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_8264_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_8265_sin__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero
thf(fact_8266_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_8267_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_8268_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_8269_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_8270_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_8271_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_8272_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_8273_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_8274_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_8275_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8276_cos__diff__cos,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8277_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8278_sin__diff__sin,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8279_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8280_sin__plus__sin,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8281_cos__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8282_cos__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8283_sin__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8284_sin__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8285_sin__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8286_sin__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8287_cos__double,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_8288_cos__double,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_8289_cos__double__sin,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_8290_cos__double__sin,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_8291_suminf__of__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ( real_V1803761363581548252l_real @ ( suminf_real @ X8 ) )
        = ( suminf_real
          @ ^ [N3: nat] : ( real_V1803761363581548252l_real @ ( X8 @ N3 ) ) ) ) ) ).

% suminf_of_real
thf(fact_8292_suminf__of__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ( real_V4546457046886955230omplex @ ( suminf_real @ X8 ) )
        = ( suminf_complex
          @ ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( X8 @ N3 ) ) ) ) ) ).

% suminf_of_real
thf(fact_8293_norm__less__p1,axiom,
    ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_8294_norm__less__p1,axiom,
    ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_8295_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_8296_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_8297_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_8298_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ( ( sin_real @ X )
            = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_8299_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_8300_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_8301_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8302_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ pi )
            & ( X
              = ( cos_real @ T3 ) )
            & ( Y
              = ( sin_real @ T3 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_8303_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8304_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_8305_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_8306_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero_02
thf(fact_8307_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8308_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_8309_cos__is__zero,axiom,
    ? [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
      & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X3 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X3 ) ) ) ).

% cos_is_zero
thf(fact_8310_norm__of__real__diff,axiom,
    ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_of_real_diff
thf(fact_8311_norm__of__real__diff,axiom,
    ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_of_real_diff
thf(fact_8312_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_8313_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_8314_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T3 ) )
              & ( Y
                = ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_8315_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T3 ) )
          & ( Y
            = ( sin_real @ T3 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8316_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T3 ) )
               => ( Y
                 != ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8317_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8318_sin__45,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_45
thf(fact_8319_cos__45,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_45
thf(fact_8320_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8321_cos__plus__cos,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8322_cos__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8323_cos__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8324_sin__gt__zero2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero2
thf(fact_8325_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8326_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_8327_sin__30,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_30
thf(fact_8328_cos__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero
thf(fact_8329_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_8330_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_8331_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_8332_cos__60,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_60
thf(fact_8333_sin__60,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_60
thf(fact_8334_cos__30,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_30
thf(fact_8335_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8336_cos__double__cos,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_8337_cos__double__cos,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_8338_cos__treble__cos,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_8339_cos__treble__cos,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_8340_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8341_sin__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_8342_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8343_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8344_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X3 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_8345_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_8346_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_8347_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X2: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_8348_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8349_sin__arctan,axiom,
    ! [X: real] :
      ( ( sin_real @ ( arctan @ X ) )
      = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_8350_cos__arctan,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_8351_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8352_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8353_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8354_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8355_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8356_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_8357_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_8358_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T3: real] :
            ( ( ord_less_real @ X @ T3 )
            & ( ord_less_real @ T3 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8359_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8360_Maclaurin__cos__expansion,axiom,
    ! [X: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8361_tan__double,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8362_tan__double,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8363_sin__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X )
        = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8364_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_8365_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_8366_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_8367_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_8368_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_8369_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_8370_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_8371_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_8372_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_8373_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_8374_tan__periodic__pi,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_pi
thf(fact_8375_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_8376_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_8377_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_8378_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_8379_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_8380_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_Suc
thf(fact_8381_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_8382_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_8383_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_8384_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_8385_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8386_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8387_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8388_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8389_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8390_tan__npi,axiom,
    ! [N2: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_8391_tan__periodic__n,axiom,
    ! [X: real,N2: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_8392_tan__periodic__nat,axiom,
    ! [X: real,N2: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_8393_tan__periodic__int,axiom,
    ! [X: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_8394_tan__periodic,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic
thf(fact_8395_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A5: complex,R3: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A5 ) ) ) ).

% complex_exp_exists
thf(fact_8396_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_8397_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_8398_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_8399_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_8400_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_8401_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_8402_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_8403_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_8404_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_8405_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_8406_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_8407_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_8408_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_8409_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_8410_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_8411_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_8412_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_mono
thf(fact_8413_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_mono
thf(fact_8414_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_mono
thf(fact_8415_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono
thf(fact_8416_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_8417_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).

% fact_dvd
thf(fact_8418_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_8419_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_8420_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8421_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8422_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8423_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8424_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8425_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8426_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8427_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8428_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8429_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_8430_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_8431_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_8432_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_8433_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_8434_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_8435_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_8436_tan__def,axiom,
    ( tan_complex
    = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X2 ) @ ( cos_complex @ X2 ) ) ) ) ).

% tan_def
thf(fact_8437_tan__def,axiom,
    ( tan_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ X2 ) @ ( cos_real @ X2 ) ) ) ) ).

% tan_def
thf(fact_8438_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% choose_dvd
thf(fact_8439_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% choose_dvd
thf(fact_8440_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% choose_dvd
thf(fact_8441_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% choose_dvd
thf(fact_8442_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% choose_dvd
thf(fact_8443_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8444_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8445_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8446_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
      = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8447_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8448_square__fact__le__2__fact,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% square_fact_le_2_fact
thf(fact_8449_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_8450_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_8451_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M5 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8452_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8453_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M5 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8454_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8455_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M5 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8456_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8457_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8458_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri5044797733671781792omplex @ N2 )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8459_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri2265585572941072030t_real @ N2 )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8460_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8461_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_8462_tan__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_gt_zero
thf(fact_8463_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_8464_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_8465_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_8466_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).

% tan_monotone
thf(fact_8467_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
thf(fact_8468_tan__minus__45,axiom,
    ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% tan_minus_45
thf(fact_8469_tan__inverse,axiom,
    ! [Y: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).

% tan_inverse
thf(fact_8470_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8471_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8472_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_8473_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_8474_tan__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_8475_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_8476_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_8477_tan__bound__pi2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_8478_tan__30,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).

% tan_30
thf(fact_8479_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X )
            = Y )
         => ( ( arctan @ Y )
            = X ) ) ) ) ).

% arctan_unique
thf(fact_8480_arctan__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X ) )
          = X ) ) ) ).

% arctan_tan
thf(fact_8481_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_8482_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > complex > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_8483_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_8484_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > rat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8485_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8486_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_8487_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
              @ ( set_ord_lessThan_nat @ N2 ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8488_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8489_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8490_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8491_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8492_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8493_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8494_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z2: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
          & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z2 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_8495_Maclaurin__exp__le,axiom,
    ! [X: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8496_tan__half,axiom,
    ( tan_complex
    = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_8497_tan__half,axiom,
    ( tan_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_8498_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8499_Maclaurin__exp__lt,axiom,
    ! [X: real,N2: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
            & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8500_cos__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_8501_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8502_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T3: real] :
          ( ( ord_less_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8503_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8504_Maclaurin__sin__expansion,axiom,
    ! [X: real,N2: nat] :
    ? [T3: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8505_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8506_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_8507_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_8508_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8509_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_8510_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_8511_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8512_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8513_sin__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( sin_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_8514_cos__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( cos_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_8515_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8516_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8517_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_8518_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_8519_sums__zero,axiom,
    ( sums_complex
    @ ^ [N3: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_8520_sums__zero,axiom,
    ( sums_real
    @ ^ [N3: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_8521_sums__zero,axiom,
    ( sums_nat
    @ ^ [N3: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_8522_sums__zero,axiom,
    ( sums_int
    @ ^ [N3: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_8523_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_8524_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_8525_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_8526_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_8527_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_8528_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_8529_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_8530_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_8531_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8532_arcsin__minus__1,axiom,
    ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arcsin_minus_1
thf(fact_8533_sums__mult,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
        @ ( times_times_complex @ C @ A ) ) ) ).

% sums_mult
thf(fact_8534_sums__mult,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
        @ ( times_times_real @ C @ A ) ) ) ).

% sums_mult
thf(fact_8535_sums__mult2,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
        @ ( times_times_complex @ A @ C ) ) ) ).

% sums_mult2
thf(fact_8536_sums__mult2,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
        @ ( times_times_real @ A @ C ) ) ) ).

% sums_mult2
thf(fact_8537_sums__diff,axiom,
    ! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
      ( ( sums_complex @ F @ A )
     => ( ( sums_complex @ G @ B )
       => ( sums_complex
          @ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% sums_diff
thf(fact_8538_sums__diff,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( minus_minus_real @ A @ B ) ) ) ) ).

% sums_diff
thf(fact_8539_sums__divide,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C )
        @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).

% sums_divide
thf(fact_8540_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_8541_sums__minus,axiom,
    ! [F: nat > real,A: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( uminus_uminus_real @ ( F @ N3 ) )
        @ ( uminus_uminus_real @ A ) ) ) ).

% sums_minus
thf(fact_8542_sums__minus,axiom,
    ! [F: nat > complex,A: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( F @ N3 ) )
        @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% sums_minus
thf(fact_8543_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% complex_diff
thf(fact_8544_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_8545_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_8546_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_8547_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8548_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8549_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8550_sums__add,axiom,
    ! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
      ( ( sums_complex @ F @ A )
     => ( ( sums_complex @ G @ B )
       => ( sums_complex
          @ ^ [N3: nat] : ( plus_plus_complex @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_complex @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8551_sums__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( sums_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8552_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8553_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8554_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8555_sums__of__real,axiom,
    ! [X8: nat > real,A: real] :
      ( ( sums_real @ X8 @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( real_V1803761363581548252l_real @ ( X8 @ N3 ) )
        @ ( real_V1803761363581548252l_real @ A ) ) ) ).

% sums_of_real
thf(fact_8556_sums__of__real,axiom,
    ! [X8: nat > real,A: real] :
      ( ( sums_real @ X8 @ A )
     => ( sums_complex
        @ ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( X8 @ N3 ) )
        @ ( real_V4546457046886955230omplex @ A ) ) ) ).

% sums_of_real
thf(fact_8557_sums__of__real__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( real_V1803761363581548252l_real @ ( F @ N3 ) )
        @ ( real_V1803761363581548252l_real @ C ) )
      = ( sums_real @ F @ C ) ) ).

% sums_of_real_iff
thf(fact_8558_sums__of__real__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( F @ N3 ) )
        @ ( real_V4546457046886955230omplex @ C ) )
      = ( sums_real @ F @ C ) ) ).

% sums_of_real_iff
thf(fact_8559_sums__sum,axiom,
    ! [I5: set_complex,F: complex > nat > real,X: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( sums_real @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_real
        @ ^ [N3: nat] :
            ( groups5808333547571424918x_real
            @ ^ [I3: complex] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups5808333547571424918x_real @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8560_sums__sum,axiom,
    ! [I5: set_real,F: real > nat > real,X: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( sums_real @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_real
        @ ^ [N3: nat] :
            ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups8097168146408367636l_real @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8561_sums__sum,axiom,
    ! [I5: set_int,F: int > nat > real,X: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( sums_real @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_real
        @ ^ [N3: nat] :
            ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups8778361861064173332t_real @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8562_sums__sum,axiom,
    ! [I5: set_real,F: real > nat > complex,X: real > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( sums_complex @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] :
            ( groups5754745047067104278omplex
            @ ^ [I3: real] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups5754745047067104278omplex @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8563_sums__sum,axiom,
    ! [I5: set_nat,F: nat > nat > complex,X: nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( sums_complex @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] :
            ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups2073611262835488442omplex @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8564_sums__sum,axiom,
    ! [I5: set_int,F: int > nat > complex,X: int > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( sums_complex @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] :
            ( groups3049146728041665814omplex
            @ ^ [I3: int] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups3049146728041665814omplex @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8565_sums__sum,axiom,
    ! [I5: set_int,F: int > nat > int,X: int > int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( sums_int @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_int
        @ ^ [N3: nat] :
            ( groups4538972089207619220nt_int
            @ ^ [I3: int] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups4538972089207619220nt_int @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8566_sums__sum,axiom,
    ! [I5: set_complex,F: complex > nat > complex,X: complex > complex] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( sums_complex @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] :
            ( groups7754918857620584856omplex
            @ ^ [I3: complex] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups7754918857620584856omplex @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8567_sums__sum,axiom,
    ! [I5: set_nat,F: nat > nat > nat,X: nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( sums_nat @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_nat
        @ ^ [N3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups3542108847815614940at_nat @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8568_sums__sum,axiom,
    ! [I5: set_nat,F: nat > nat > real,X: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( sums_real @ ( F @ I2 ) @ ( X @ I2 ) ) )
     => ( sums_real
        @ ^ [N3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( F @ I3 @ N3 )
            @ I5 )
        @ ( groups6591440286371151544t_real @ X @ I5 ) ) ) ).

% sums_sum
thf(fact_8569_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8570_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8571_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8572_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8573_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_8574_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% complex_add
thf(fact_8575_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8576_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8577_sums__Suc__imp,axiom,
    ! [F: nat > complex,S: complex] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
          @ S )
       => ( sums_complex @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_8578_sums__Suc__imp,axiom,
    ! [F: nat > real,S: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
          @ S )
       => ( sums_real @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_8579_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8580_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8581_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8582_sums__Suc,axiom,
    ! [F: nat > complex,L: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_complex @ F @ ( plus_plus_complex @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8583_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8584_sums__Suc__iff,axiom,
    ! [F: nat > complex,S: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ S )
      = ( sums_complex @ F @ ( plus_plus_complex @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8585_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > complex,S: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ( F @ I2 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_complex @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_8586_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ( F @ I2 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_8587_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_8588_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_complex
        @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8589_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8590_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8591_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8592_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_complex
        @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8593_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8594_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8595_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8596_sums__finite,axiom,
    ! [N5: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_complex ) )
       => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N5 ) ) ) ) ).

% sums_finite
thf(fact_8597_sums__finite,axiom,
    ! [N5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N5 ) ) ) ) ).

% sums_finite
thf(fact_8598_sums__finite,axiom,
    ! [N5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N5 ) ) ) ) ).

% sums_finite
thf(fact_8599_sums__finite,axiom,
    ! [N5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N: nat] :
            ( ~ ( member_nat @ N @ N5 )
           => ( ( F @ N )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N5 ) ) ) ) ).

% sums_finite
thf(fact_8600_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_8601_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_8602_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_8603_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y @ R2 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_8604_complex__of__real__add__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( plus_plus_real @ R2 @ X ) @ Y ) ) ).

% complex_of_real_add_Complex
thf(fact_8605_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( plus_plus_real @ X @ R2 ) @ Y ) ) ).

% Complex_add_complex_of_real
thf(fact_8606_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_8607_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_8608_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_8609_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_8610_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_8611_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_8612_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_8613_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N3 ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8614_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N3 ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8615_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N3 ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8616_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8617_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8618_sums__iff__shift,axiom,
    ! [F: nat > complex,N2: nat,S: complex] :
      ( ( sums_complex
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ S )
      = ( sums_complex @ F @ ( plus_plus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_8619_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_8620_sums__iff__shift_H,axiom,
    ! [F: nat > complex,N2: nat,S: complex] :
      ( ( sums_complex
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_complex @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_8621_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_8622_sums__split__initial__segment,axiom,
    ! [F: nat > complex,S: complex,N2: nat] :
      ( ( sums_complex @ F @ S )
     => ( sums_complex
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_8623_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N2: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_8624_sums__If__finite__set_H,axiom,
    ! [G: nat > complex,S3: complex,A2: set_nat,S5: complex,F: nat > complex] :
      ( ( sums_complex @ G @ S3 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S5
            = ( plus_plus_complex @ S3
              @ ( groups2073611262835488442omplex
                @ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) )
                @ A2 ) ) )
         => ( sums_complex
            @ ^ [N3: nat] : ( if_complex @ ( member_nat @ N3 @ A2 ) @ ( F @ N3 ) @ ( G @ N3 ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_8625_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A2: set_nat,S5: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S5
            = ( plus_plus_real @ S3
              @ ( groups6591440286371151544t_real
                @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
                @ A2 ) ) )
         => ( sums_real
            @ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A2 ) @ ( F @ N3 ) @ ( G @ N3 ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_8626_Complex__sum_H,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [X2: nat] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
        @ S )
      = ( complex2 @ ( groups6591440286371151544t_real @ F @ S ) @ zero_zero_real ) ) ).

% Complex_sum'
thf(fact_8627_Complex__sum_H,axiom,
    ! [F: complex > real,S: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
        @ S )
      = ( complex2 @ ( groups5808333547571424918x_real @ F @ S ) @ zero_zero_real ) ) ).

% Complex_sum'
thf(fact_8628_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_8629_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% complex_mult
thf(fact_8630_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_8631_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8632_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_8633_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_8634_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_8635_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8636_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_8637_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_8638_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_8639_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_8640_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8641_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_8642_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_8643_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_8644_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_8645_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_8646_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_8647_power__half__series,axiom,
    ( sums_real
    @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8648_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_8649_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_8650_complex__norm,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_8651_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_8652_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_8653_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8654_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_8655_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_8656_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8657_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8658_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_8659_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_8660_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_8661_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_8662_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8663_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8664_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_8665_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8666_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8667_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_8668_diffs__equiv,axiom,
    ! [C: nat > real,X: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8669_diffs__equiv,axiom,
    ! [C: nat > complex,X: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8670_monoI1,axiom,
    ! [X8: nat > real] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_real @ ( X8 @ M3 ) @ ( X8 @ N ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% monoI1
thf(fact_8671_monoI1,axiom,
    ! [X8: nat > set_nat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_set_nat @ ( X8 @ M3 ) @ ( X8 @ N ) ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% monoI1
thf(fact_8672_monoI1,axiom,
    ! [X8: nat > rat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_rat @ ( X8 @ M3 ) @ ( X8 @ N ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% monoI1
thf(fact_8673_monoI1,axiom,
    ! [X8: nat > num] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_num @ ( X8 @ M3 ) @ ( X8 @ N ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% monoI1
thf(fact_8674_monoI1,axiom,
    ! [X8: nat > nat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_nat @ ( X8 @ M3 ) @ ( X8 @ N ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% monoI1
thf(fact_8675_monoI1,axiom,
    ! [X8: nat > int] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_int @ ( X8 @ M3 ) @ ( X8 @ N ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% monoI1
thf(fact_8676_monoI2,axiom,
    ! [X8: nat > real] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_real @ ( X8 @ N ) @ ( X8 @ M3 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% monoI2
thf(fact_8677_monoI2,axiom,
    ! [X8: nat > set_nat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_set_nat @ ( X8 @ N ) @ ( X8 @ M3 ) ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% monoI2
thf(fact_8678_monoI2,axiom,
    ! [X8: nat > rat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_rat @ ( X8 @ N ) @ ( X8 @ M3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% monoI2
thf(fact_8679_monoI2,axiom,
    ! [X8: nat > num] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_num @ ( X8 @ N ) @ ( X8 @ M3 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% monoI2
thf(fact_8680_monoI2,axiom,
    ! [X8: nat > nat] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_nat @ ( X8 @ N ) @ ( X8 @ M3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% monoI2
thf(fact_8681_monoI2,axiom,
    ! [X8: nat > int] :
      ( ! [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
         => ( ord_less_eq_int @ ( X8 @ N ) @ ( X8 @ M3 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% monoI2
thf(fact_8682_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X4: nat > real] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_real @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_real @ ( X4 @ N3 ) @ ( X4 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8683_monoseq__def,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X4: nat > set_nat] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_set_nat @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_set_nat @ ( X4 @ N3 ) @ ( X4 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8684_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X4: nat > rat] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_rat @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_rat @ ( X4 @ N3 ) @ ( X4 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8685_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X4: nat > num] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_num @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_num @ ( X4 @ N3 ) @ ( X4 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8686_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X4: nat > nat] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_nat @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_nat @ ( X4 @ N3 ) @ ( X4 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8687_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X4: nat > int] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_int @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_int @ ( X4 @ N3 ) @ ( X4 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8688_diffs__of__real,axiom,
    ! [F: nat > real] :
      ( ( diffs_complex
        @ ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( F @ N3 ) ) )
      = ( ^ [N3: nat] : ( real_V4546457046886955230omplex @ ( diffs_real @ F @ N3 ) ) ) ) ).

% diffs_of_real
thf(fact_8689_diffs__minus,axiom,
    ! [C: nat > int] :
      ( ( diffs_int
        @ ^ [N3: nat] : ( uminus_uminus_int @ ( C @ N3 ) ) )
      = ( ^ [N3: nat] : ( uminus_uminus_int @ ( diffs_int @ C @ N3 ) ) ) ) ).

% diffs_minus
thf(fact_8690_diffs__minus,axiom,
    ! [C: nat > real] :
      ( ( diffs_real
        @ ^ [N3: nat] : ( uminus_uminus_real @ ( C @ N3 ) ) )
      = ( ^ [N3: nat] : ( uminus_uminus_real @ ( diffs_real @ C @ N3 ) ) ) ) ).

% diffs_minus
thf(fact_8691_diffs__minus,axiom,
    ! [C: nat > complex] :
      ( ( diffs_complex
        @ ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( C @ N3 ) ) )
      = ( ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( diffs_complex @ C @ N3 ) ) ) ) ).

% diffs_minus
thf(fact_8692_diffs__minus,axiom,
    ! [C: nat > rat] :
      ( ( diffs_rat
        @ ^ [N3: nat] : ( uminus_uminus_rat @ ( C @ N3 ) ) )
      = ( ^ [N3: nat] : ( uminus_uminus_rat @ ( diffs_rat @ C @ N3 ) ) ) ) ).

% diffs_minus
thf(fact_8693_diffs__minus,axiom,
    ! [C: nat > code_integer] :
      ( ( diffs_Code_integer
        @ ^ [N3: nat] : ( uminus1351360451143612070nteger @ ( C @ N3 ) ) )
      = ( ^ [N3: nat] : ( uminus1351360451143612070nteger @ ( diffs_Code_integer @ C @ N3 ) ) ) ) ).

% diffs_minus
thf(fact_8694_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C3: nat > rat,N3: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8695_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C3: nat > int,N3: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8696_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C3: nat > real,N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8697_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C3: nat > complex,N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8698_termdiff__converges__all,axiom,
    ! [C: nat > complex,X: complex] :
      ( ! [X3: complex] :
          ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% termdiff_converges_all
thf(fact_8699_termdiff__converges__all,axiom,
    ! [C: nat > real,X: real] :
      ( ! [X3: real] :
          ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% termdiff_converges_all
thf(fact_8700_termdiff__converges,axiom,
    ! [X: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
           => ( summable_real
              @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8701_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
     => ( ! [X3: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
           => ( summable_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8702_monoseq__minus,axiom,
    ! [A: nat > int] :
      ( ( topolo4899668324122417113eq_int @ A )
     => ( topolo4899668324122417113eq_int
        @ ^ [N3: nat] : ( uminus_uminus_int @ ( A @ N3 ) ) ) ) ).

% monoseq_minus
thf(fact_8703_monoseq__minus,axiom,
    ! [A: nat > rat] :
      ( ( topolo4267028734544971653eq_rat @ A )
     => ( topolo4267028734544971653eq_rat
        @ ^ [N3: nat] : ( uminus_uminus_rat @ ( A @ N3 ) ) ) ) ).

% monoseq_minus
thf(fact_8704_monoseq__minus,axiom,
    ! [A: nat > code_integer] :
      ( ( topolo2919662092509805066nteger @ A )
     => ( topolo2919662092509805066nteger
        @ ^ [N3: nat] : ( uminus1351360451143612070nteger @ ( A @ N3 ) ) ) ) ).

% monoseq_minus
thf(fact_8705_monoseq__minus,axiom,
    ! [A: nat > real] :
      ( ( topolo6980174941875973593q_real @ A )
     => ( topolo6980174941875973593q_real
        @ ^ [N3: nat] : ( uminus_uminus_real @ ( A @ N3 ) ) ) ) ).

% monoseq_minus
thf(fact_8706_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X4: nat > real] :
          ( ! [N3: nat] : ( ord_less_eq_real @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_real @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8707_monoseq__Suc,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X4: nat > set_nat] :
          ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_set_nat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8708_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X4: nat > rat] :
          ( ! [N3: nat] : ( ord_less_eq_rat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_rat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8709_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X4: nat > num] :
          ( ! [N3: nat] : ( ord_less_eq_num @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_num @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8710_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X4: nat > nat] :
          ( ! [N3: nat] : ( ord_less_eq_nat @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_nat @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8711_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X4: nat > int] :
          ( ! [N3: nat] : ( ord_less_eq_int @ ( X4 @ N3 ) @ ( X4 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_int @ ( X4 @ ( suc @ N3 ) ) @ ( X4 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8712_mono__SucI2,axiom,
    ! [X8: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% mono_SucI2
thf(fact_8713_mono__SucI2,axiom,
    ! [X8: nat > set_nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% mono_SucI2
thf(fact_8714_mono__SucI2,axiom,
    ! [X8: nat > rat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% mono_SucI2
thf(fact_8715_mono__SucI2,axiom,
    ! [X8: nat > num] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% mono_SucI2
thf(fact_8716_mono__SucI2,axiom,
    ! [X8: nat > nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% mono_SucI2
thf(fact_8717_mono__SucI2,axiom,
    ! [X8: nat > int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% mono_SucI2
thf(fact_8718_mono__SucI1,axiom,
    ! [X8: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% mono_SucI1
thf(fact_8719_mono__SucI1,axiom,
    ! [X8: nat > set_nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% mono_SucI1
thf(fact_8720_mono__SucI1,axiom,
    ! [X8: nat > rat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% mono_SucI1
thf(fact_8721_mono__SucI1,axiom,
    ! [X8: nat > num] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% mono_SucI1
thf(fact_8722_mono__SucI1,axiom,
    ! [X8: nat > nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% mono_SucI1
thf(fact_8723_mono__SucI1,axiom,
    ! [X8: nat > int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% mono_SucI1
thf(fact_8724_exp__two__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
    = one_one_complex ) ).

% exp_two_pi_i'
thf(fact_8725_exp__two__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
    = one_one_complex ) ).

% exp_two_pi_i
thf(fact_8726_pochhammer__double,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8727_pochhammer__double,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8728_pochhammer__double,axiom,
    ! [Z: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8729_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8730_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_real @ N3 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_8731_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_rat @ N3 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_8732_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_8733_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_8734_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_8735_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_8736_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_8737_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_8738_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_8739_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_8740_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_8741_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_8742_complex__i__mult__minus,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% complex_i_mult_minus
thf(fact_8743_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_8744_floor__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_8745_floor__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_8746_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_8747_divide__numeral__i,axiom,
    ! [Z: complex,N2: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N2 ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% divide_numeral_i
thf(fact_8748_zero__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_floor
thf(fact_8749_zero__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).

% zero_le_floor
thf(fact_8750_floor__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_8751_floor__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_8752_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_8753_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_8754_zero__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% zero_less_floor
thf(fact_8755_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_8756_floor__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_le_zero
thf(fact_8757_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_8758_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_8759_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_8760_one__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% one_le_floor
thf(fact_8761_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_8762_floor__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_less_one
thf(fact_8763_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_8764_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_8765_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_8766_floor__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_8767_floor__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_8768_floor__diff__one,axiom,
    ! [X: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_8769_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_8770_floor__numeral__power,axiom,
    ! [X: num,N2: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).

% floor_numeral_power
thf(fact_8771_floor__numeral__power,axiom,
    ! [X: num,N2: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ).

% floor_numeral_power
thf(fact_8772_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_8773_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_less_floor
thf(fact_8774_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_8775_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_8776_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_8777_one__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_8778_one__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_8779_floor__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8780_floor__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8781_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_8782_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_8783_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8784_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8785_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_8786_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_8787_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_8788_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_8789_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_8790_i__even__power,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) ) ).

% i_even_power
thf(fact_8791_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_8792_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_8793_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_8794_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_8795_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_8796_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_8797_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_8798_of__int__floor__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_8799_of__int__floor__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_8800_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_8801_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_8802_pochhammer__pos,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8803_pochhammer__pos,axiom,
    ! [X: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8804_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8805_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8806_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ M )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8807_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8808_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8809_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ N2 )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8810_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N2 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8811_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8812_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_8813_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_8814_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_8815_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_8816_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_8817_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_8818_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_8819_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_8820_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_8821_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_8822_le__floor__add,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_8823_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_8824_int__add__floor,axiom,
    ! [Z: int,X: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_8825_int__add__floor,axiom,
    ! [Z: int,X: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_8826_floor__add__int,axiom,
    ! [X: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_8827_floor__add__int,axiom,
    ! [X: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_8828_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_8829_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_8830_floor__power,axiom,
    ! [X: real,N2: nat] :
      ( ( X
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N2 ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N2 ) ) ) ).

% floor_power
thf(fact_8831_floor__power,axiom,
    ! [X: rat,N2: nat] :
      ( ( X
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N2 ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N2 ) ) ) ).

% floor_power
thf(fact_8832_pochhammer__nonneg,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8833_pochhammer__nonneg,axiom,
    ! [X: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8834_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8835_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8836_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_8837_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_8838_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_8839_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_8840_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_8841_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_8842_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( ( complex2 @ X @ Y )
        = imaginary_unit )
      = ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_8843_one__add__floor,axiom,
    ! [X: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_8844_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_8845_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_8846_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_8847_i__mult__Complex,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% i_mult_Complex
thf(fact_8848_Complex__mult__i,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% Complex_mult_i
thf(fact_8849_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_8850_floor__eq,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N2 ) ) ) ).

% floor_eq
thf(fact_8851_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_8852_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_8853_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_8854_pochhammer__rec,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8855_pochhammer__rec,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8856_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8857_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8858_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8859_pochhammer__rec_H,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8860_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8861_pochhammer__rec_H,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8862_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8863_pochhammer__rec_H,axiom,
    ! [Z: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8864_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8865_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8866_pochhammer__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8867_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8868_pochhammer__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8869_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8870_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8871_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8872_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8873_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8874_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8875_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8876_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8877_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8878_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8879_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8880_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8881_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8882_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8883_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8884_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8885_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8886_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8887_pochhammer__product_H,axiom,
    ! [Z: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8888_pochhammer__product_H,axiom,
    ! [Z: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8889_pochhammer__product_H,axiom,
    ! [Z: real,N2: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8890_pochhammer__product_H,axiom,
    ! [Z: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8891_pochhammer__product_H,axiom,
    ! [Z: complex,N2: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8892_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_8893_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_8894_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_8895_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_8896_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_8897_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_8898_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8899_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8900_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% less_floor_iff
thf(fact_8901_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_8902_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_8903_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_8904_floor__correct,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8905_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8906_floor__eq2,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_8907_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_8908_complex__of__real__i,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% complex_of_real_i
thf(fact_8909_i__complex__of__real,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% i_complex_of_real
thf(fact_8910_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8911_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8912_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8913_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8914_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s2602460028002588243omplex @ Z @ N2 )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8915_Complex__eq,axiom,
    ( complex2
    = ( ^ [A3: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).

% Complex_eq
thf(fact_8916_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_8917_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_8918_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8919_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8920_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8921_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8922_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8923_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_8924_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_8925_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% pochhammer_same
thf(fact_8926_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% pochhammer_same
thf(fact_8927_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% pochhammer_same
thf(fact_8928_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% pochhammer_same
thf(fact_8929_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% pochhammer_same
thf(fact_8930_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A5: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_8931_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8932_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8933_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8934_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8935_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8936_pochhammer__minus,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8937_pochhammer__minus,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8938_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8939_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8940_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8941_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8942_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8943_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_8944_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_8945_cmod__complex__polar,axiom,
    ! [R2: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R2 ) ) ).

% cmod_complex_polar
thf(fact_8946_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_8947_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_double
thf(fact_8948_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_double
thf(fact_8949_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_double
thf(fact_8950_floor__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8951_Arg__minus__ii,axiom,
    ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
    = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_minus_ii
thf(fact_8952_csqrt__ii,axiom,
    ( ( csqrt @ imaginary_unit )
    = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt_ii
thf(fact_8953_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8954_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8955_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N2: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8956_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A3: rat,N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_8957_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A3: int,N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_8958_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A3: real,N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_8959_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A3: complex,N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_8960_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A3: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_8961_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_8962_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_8963_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_8964_of__nat__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8965_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8966_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8967_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8968_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8969_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8970_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X2: nat] : ( ring_1_of_int_rat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8971_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( ring_1_of_int_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8972_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups2316167850115554303t_real
        @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8973_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1072433553688619179nt_rat
        @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8974_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8975_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_8976_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_8977_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_8978_prod_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
      = one_one_complex ) ).

% prod.empty
thf(fact_8979_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_8980_prod_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
      = one_one_rat ) ).

% prod.empty
thf(fact_8981_prod_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
      = one_one_nat ) ).

% prod.empty
thf(fact_8982_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_8983_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_8984_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_8985_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_8986_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_8987_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_8988_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8989_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8990_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8991_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8992_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8993_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8994_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_8995_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8996_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8997_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8998_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8999_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_9000_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_9001_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_9002_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_9003_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_9004_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_9005_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_9006_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_9007_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_9008_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_9009_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_9010_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_9011_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_9012_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_9013_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_9014_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_9015_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9016_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9017_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9018_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9019_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9020_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9021_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9022_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9023_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9024_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_9025_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_9026_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_9027_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_9028_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_9029_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9030_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9031_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9032_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9033_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9034_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9035_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9036_prod_Oswap,axiom,
    ! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% prod.swap
thf(fact_9037_prod_Oswap,axiom,
    ! [G: nat > nat > int,B3: set_nat,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% prod.swap
thf(fact_9038_prod_Oswap,axiom,
    ! [G: nat > int > int,B3: set_int,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups1705073143266064639nt_int @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups1705073143266064639nt_int
        @ ^ [J3: int] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% prod.swap
thf(fact_9039_prod_Oswap,axiom,
    ! [G: int > nat > int,B3: set_nat,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I3: int] : ( groups705719431365010083at_int @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups1705073143266064639nt_int
            @ ^ [I3: int] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% prod.swap
thf(fact_9040_prod_Oswap,axiom,
    ! [G: int > int > int,B3: set_int,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [I3: int] : ( groups1705073143266064639nt_int @ ( G @ I3 ) @ B3 )
        @ A2 )
      = ( groups1705073143266064639nt_int
        @ ^ [J3: int] :
            ( groups1705073143266064639nt_int
            @ ^ [I3: int] : ( G @ I3 @ J3 )
            @ A2 )
        @ B3 ) ) ).

% prod.swap
thf(fact_9041_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_9042_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_9043_prod_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_9044_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > complex,A2: set_complex] :
      ( ( ( groups3708469109370488835omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9045_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups713298508707869441omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9046_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9047_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups7440179247065528705omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9048_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A2: set_complex] :
      ( ( ( groups766887009212190081x_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9049_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9050_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A2: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9051_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9052_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > rat,A2: set_complex] :
      ( ( ( groups225925009352817453ex_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9053_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups4061424788464935467al_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9054_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_9055_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_9056_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_9057_prod__power__distrib,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
      = ( groups708209901874060359at_nat
        @ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_9058_prod__power__distrib,axiom,
    ! [F: nat > int,A2: set_nat,N2: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_9059_prod__power__distrib,axiom,
    ! [F: int > int,A2: set_int,N2: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_9060_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups4696554848551431203al_nat
            @ ^ [X2: real] :
                ( groups708209901874060359at_nat @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups4696554848551431203al_nat
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9061_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups861055069439313189ex_nat
            @ ^ [X2: complex] :
                ( groups708209901874060359at_nat @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups861055069439313189ex_nat
                @ ^ [X2: complex] : ( G @ X2 @ Y2 )
                @ ( collect_complex
                  @ ^ [X2: complex] :
                      ( ( member_complex @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9062_prod_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups1707563613775114915nt_nat
            @ ^ [X2: int] :
                ( groups708209901874060359at_nat @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups708209901874060359at_nat
            @ ^ [Y2: nat] :
                ( groups1707563613775114915nt_nat
                @ ^ [X2: int] : ( G @ X2 @ Y2 )
                @ ( collect_int
                  @ ^ [X2: int] :
                      ( ( member_int @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9063_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: real > nat > int,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups4694064378042380927al_int
            @ ^ [X2: real] :
                ( groups705719431365010083at_int @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y2: nat] :
                ( groups4694064378042380927al_int
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9064_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_nat,G: complex > nat > int,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups858564598930262913ex_int
            @ ^ [X2: complex] :
                ( groups705719431365010083at_int @ ( G @ X2 )
                @ ( collect_nat
                  @ ^ [Y2: nat] :
                      ( ( member_nat @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups705719431365010083at_int
            @ ^ [Y2: nat] :
                ( groups858564598930262913ex_int
                @ ^ [X2: complex] : ( G @ X2 @ Y2 )
                @ ( collect_complex
                  @ ^ [X2: complex] :
                      ( ( member_complex @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9065_prod_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_int,G: real > int > int,R: real > int > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups4694064378042380927al_int
            @ ^ [X2: real] :
                ( groups1705073143266064639nt_int @ ( G @ X2 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y2: int] :
                ( groups4694064378042380927al_int
                @ ^ [X2: real] : ( G @ X2 @ Y2 )
                @ ( collect_real
                  @ ^ [X2: real] :
                      ( ( member_real @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9066_prod_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_int,G: complex > int > int,R: complex > int > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups858564598930262913ex_int
            @ ^ [X2: complex] :
                ( groups1705073143266064639nt_int @ ( G @ X2 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups1705073143266064639nt_int
            @ ^ [Y2: int] :
                ( groups858564598930262913ex_int
                @ ^ [X2: complex] : ( G @ X2 @ Y2 )
                @ ( collect_complex
                  @ ^ [X2: complex] :
                      ( ( member_complex @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9067_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_real,G: nat > real > nat,R: nat > real > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_real @ B3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X2: nat] :
                ( groups4696554848551431203al_nat @ ( G @ X2 )
                @ ( collect_real
                  @ ^ [Y2: real] :
                      ( ( member_real @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups4696554848551431203al_nat
            @ ^ [Y2: real] :
                ( groups708209901874060359at_nat
                @ ^ [X2: nat] : ( G @ X2 @ Y2 )
                @ ( collect_nat
                  @ ^ [X2: nat] :
                      ( ( member_nat @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9068_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_complex,G: nat > complex > nat,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X2: nat] :
                ( groups861055069439313189ex_nat @ ( G @ X2 )
                @ ( collect_complex
                  @ ^ [Y2: complex] :
                      ( ( member_complex @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups861055069439313189ex_nat
            @ ^ [Y2: complex] :
                ( groups708209901874060359at_nat
                @ ^ [X2: nat] : ( G @ X2 @ Y2 )
                @ ( collect_nat
                  @ ^ [X2: nat] :
                      ( ( member_nat @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9069_prod_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_int,G: nat > int > nat,R: nat > int > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B3 )
       => ( ( groups708209901874060359at_nat
            @ ^ [X2: nat] :
                ( groups1707563613775114915nt_nat @ ( G @ X2 )
                @ ( collect_int
                  @ ^ [Y2: int] :
                      ( ( member_int @ Y2 @ B3 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ A2 )
          = ( groups1707563613775114915nt_nat
            @ ^ [Y2: int] :
                ( groups708209901874060359at_nat
                @ ^ [X2: nat] : ( G @ X2 @ Y2 )
                @ ( collect_nat
                  @ ^ [X2: nat] :
                      ( ( member_nat @ X2 @ A2 )
                      & ( R @ X2 @ Y2 ) ) ) )
            @ B3 ) ) ) ) ).

% prod.swap_restrict
thf(fact_9070_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_9071_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A2: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_9072_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_9073_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_9074_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_9075_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_9076_prod__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9077_prod__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9078_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9079_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9080_prod__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9081_prod__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9082_prod__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9083_prod__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9084_prod__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9085_prod__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9086_prod__pos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_9087_prod__pos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_9088_prod__pos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_9089_prod__ge__1,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9090_prod__ge__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9091_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9092_prod__ge__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9093_prod__ge__1,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9094_prod__ge__1,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9095_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9096_prod__ge__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9097_prod__ge__1,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9098_prod__ge__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_9099_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A3: nat] : ( times_times_complex @ ( F @ A3 ) )
        @ A
        @ B
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_9100_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A3: nat] : ( times_times_real @ ( F @ A3 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_9101_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A3: nat] : ( times_times_rat @ ( F @ A3 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_9102_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A3: nat] : ( times_times_nat @ ( F @ A3 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_9103_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A3: nat] : ( times_times_int @ ( F @ A3 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_9104_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups713298508707869441omplex @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups713298508707869441omplex
          @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9105_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups6464643781859351333omplex
          @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9106_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups3708469109370488835omplex
          @ ^ [X2: complex] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9107_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups7440179247065528705omplex
          @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9108_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups1681761925125756287l_real
          @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9109_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9110_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups766887009212190081x_real
          @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9111_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9112_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups4061424788464935467al_rat
          @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9113_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups73079841787564623at_rat
          @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_9114_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_9115_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_9116_power__sum,axiom,
    ! [C: real,F: nat > nat,A2: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [A3: nat] : ( power_power_real @ C @ ( F @ A3 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_9117_power__sum,axiom,
    ! [C: complex,F: nat > nat,A2: set_nat] :
      ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [A3: nat] : ( power_power_complex @ C @ ( F @ A3 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_9118_power__sum,axiom,
    ! [C: nat,F: nat > nat,A2: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A3: nat] : ( power_power_nat @ C @ ( F @ A3 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_9119_power__sum,axiom,
    ! [C: int,F: nat > nat,A2: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [A3: nat] : ( power_power_int @ C @ ( F @ A3 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_9120_power__sum,axiom,
    ! [C: int,F: int > nat,A2: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A3: int] : ( power_power_int @ C @ ( F @ A3 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_9121_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_9122_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_9123_prod__le__1,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_9124_prod__le__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_9125_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_9126_prod__le__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_9127_prod__le__1,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_9128_prod__le__1,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_9129_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_9130_prod__le__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_9131_prod__le__1,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_9132_prod__le__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_9133_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2316167850115554303t_real @ H2 @ S3 ) @ ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9134_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups73079841787564623at_rat @ H2 @ S3 ) @ ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9135_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups225925009352817453ex_rat @ H2 @ S3 ) @ ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9136_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1072433553688619179nt_rat @ H2 @ S3 ) @ ( groups1072433553688619179nt_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9137_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups861055069439313189ex_nat @ H2 @ S3 ) @ ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9138_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1707563613775114915nt_nat @ H2 @ S3 ) @ ( groups1707563613775114915nt_nat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9139_prod_Orelated,axiom,
    ! [R: int > int > $o,S3: set_complex,H2: complex > int,G: complex > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_int @ X16 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups858564598930262913ex_int @ H2 @ S3 ) @ ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9140_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_nat,H2: nat > nat,G: nat > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups708209901874060359at_nat @ H2 @ S3 ) @ ( groups708209901874060359at_nat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9141_prod_Orelated,axiom,
    ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_int @ X16 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups705719431365010083at_int @ H2 @ S3 ) @ ( groups705719431365010083at_int @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9142_prod_Orelated,axiom,
    ! [R: int > int > $o,S3: set_int,H2: int > int,G: int > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_int @ X16 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1705073143266064639nt_int @ H2 @ S3 ) @ ( groups1705073143266064639nt_int @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_9143_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9144_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_9145_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9146_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_9147_cis__minus__pi__half,axiom,
    ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
    = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).

% cis_minus_pi_half
thf(fact_9148_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_9149_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9150_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9151_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9152_cis__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).

% cis_divide
thf(fact_9153_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_9154_DeMoivre,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N2 )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) ) ) ).

% DeMoivre
thf(fact_9155_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9156_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B2: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_9157_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9158_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K2 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).

% binomial_code
thf(fact_9159_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_9160_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_9161_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_9162_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_9163_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_9164_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_9165_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_9166_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq_nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_9167_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_9168_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ K2 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9169_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_9170_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9171_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9172_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_9173_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_9174_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_9175_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_9176_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_9177_binomial__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_9178_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9179_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9180_vandermonde,axiom,
    ! [M: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_9181_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9182_choose__row__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% choose_row_sum
thf(fact_9183_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_9184_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_9185_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_9186_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_9187_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_9188_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_9189_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N2 @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% choose_square_sum
thf(fact_9190_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9191_binomial__absorption,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_9192_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_9193_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9194_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_9195_binomial__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_9196_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% binomial_maximum'
thf(fact_9197_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9198_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9199_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_9200_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9201_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9202_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_9203_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9204_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9205_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9206_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9207_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_9208_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9209_choose__two,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_9210_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9211_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_9212_Maclaurin__sin__bound,axiom,
    ! [X: real,N2: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N2 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9213_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] : N3 ) ) ).

% of_nat_id
thf(fact_9214_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_9215_real__sqrt__inverse,axiom,
    ! [X: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_inverse
thf(fact_9216_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X2: real,Y2: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).

% divide_real_def
thf(fact_9217_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).

% complex_scaleR
thf(fact_9218_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_9219_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9220_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_9221_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N3: nat] :
            ( ( N3 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_9222_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9223_ln__inverse,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).

% ln_inverse
thf(fact_9224_log__inverse,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_9225_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9226_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9227_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9228_tan__cot,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).

% tan_cot
thf(fact_9229_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9230_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9231_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9232_cot__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9233_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% sinh_real_less_iff
thf(fact_9234_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% sinh_real_le_iff
thf(fact_9235_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% sinh_real_pos_iff
thf(fact_9236_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9237_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9238_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_9239_real__root__zero,axiom,
    ! [N2: nat] :
      ( ( root @ N2 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9240_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9241_real__root__eq__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = ( root @ N2 @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9242_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9243_real__root__eq__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9244_real__root__less__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9245_real__root__le__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9246_real__root__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9247_real__root__eq__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9248_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9249_real__root__lt__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9250_real__root__le__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9251_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9252_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9253_real__root__lt__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9254_real__root__le__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9255_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9256_cot__npi,axiom,
    ! [N2: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9257_real__root__pow__pos2,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9258_cot__periodic,axiom,
    ! [X: real] :
      ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X ) ) ).

% cot_periodic
thf(fact_9259_real__root__inverse,axiom,
    ! [N2: nat,X: real] :
      ( ( root @ N2 @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( root @ N2 @ X ) ) ) ).

% real_root_inverse
thf(fact_9260_real__root__commute,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( root @ M @ ( root @ N2 @ X ) )
      = ( root @ N2 @ ( root @ M @ X ) ) ) ).

% real_root_commute
thf(fact_9261_sinh__le__cosh__real,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_le_cosh_real
thf(fact_9262_sinh__less__cosh__real,axiom,
    ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_less_cosh_real
thf(fact_9263_real__root__mult,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( root @ N2 @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).

% real_root_mult
thf(fact_9264_real__root__mult__exp,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N2 ) @ X )
      = ( root @ M @ ( root @ N2 @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9265_real__root__minus,axiom,
    ! [N2: nat,X: real] :
      ( ( root @ N2 @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( root @ N2 @ X ) ) ) ).

% real_root_minus
thf(fact_9266_real__root__divide,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( root @ N2 @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).

% real_root_divide
thf(fact_9267_real__root__pos__pos__le,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9268_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_9269_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9270_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9271_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9272_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_9273_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X2: complex,Y2: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y2 ) ) ) ) ).

% divide_complex_def
thf(fact_9274_real__root__less__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9275_real__root__le__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9276_real__root__power,axiom,
    ! [N2: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9277_real__root__abs,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_abs
thf(fact_9278_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9279_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9280_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9281_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_9282_real__root__gt__zero,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9283_real__root__strict__decreasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9284_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9285_root__abs__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9286_real__root__pos__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9287_real__root__strict__increasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9288_real__root__decreasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9289_real__root__pow__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9290_real__root__power__cancel,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9291_real__root__pos__unique,axiom,
    ! [N2: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N2 )
            = X )
         => ( ( root @ N2 @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9292_odd__real__root__pow,axiom,
    ! [N2: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9293_odd__real__root__unique,axiom,
    ! [N2: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ( power_power_real @ Y @ N2 )
          = X )
       => ( ( root @ N2 @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9294_odd__real__root__power__cancel,axiom,
    ! [N2: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9295_real__root__increasing,axiom,
    ! [N2: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9296_log__root,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N2 @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_9297_log__base__root,axiom,
    ! [N2: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9298_ln__root,axiom,
    ! [N2: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N2 @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_9299_root__powr__inverse,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9300_complex__inverse,axiom,
    ! [A: real,B: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_9301_cosh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( cosh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9302_cot__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).

% cot_gt_zero
thf(fact_9303_sinh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( sinh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9304_tan__cot_H,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( cot_real @ X ) ) ).

% tan_cot'
thf(fact_9305_arctan__def,axiom,
    ( arctan
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arctan_def
thf(fact_9306_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arcsin_def
thf(fact_9307_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9308_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9309_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9310_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9311_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9312_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9313_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9314_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).

% signed_take_bit_negative_iff
thf(fact_9315_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9316_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9317_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9318_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9319_bit__and__int__iff,axiom,
    ! [K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        & ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).

% bit_and_int_iff
thf(fact_9320_bit__or__int__iff,axiom,
    ! [K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        | ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).

% bit_or_int_iff
thf(fact_9321_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% int_sgnE
thf(fact_9322_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L ) )
     => ( ( divide_divide_int @ K @ L )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_9323_ln__real__def,axiom,
    ( ln_ln_real
    = ( ^ [X2: real] :
          ( the_real
          @ ^ [U2: real] :
              ( ( exp_real @ U2 )
              = X2 ) ) ) ) ).

% ln_real_def
thf(fact_9324_bit__not__int__iff_H,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N2 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_not_int_iff'
thf(fact_9325_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_9326_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X2: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9327_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9328_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9329_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9330_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9331_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K2: int] : ( bit_concat_bit @ N3 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9332_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ N @ M4 )
             => ( ( bit_se1146084159140164899it_int @ K @ M4 )
                = ( bit_se1146084159140164899it_int @ K @ N ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ) ) ).

% int_bit_bound
thf(fact_9333_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K2: int,N3: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_int_def
thf(fact_9334_arccos__def,axiom,
    ( arccos
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X2 )
              & ( ord_less_eq_real @ X2 @ pi )
              & ( ( cos_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arccos_def
thf(fact_9335_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9336_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K2: int] :
              ( ( A12 = K2 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
          | ? [L2: int,K2: int,Q4: int] :
              ( ( A12 = K2 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K2
                = ( times_times_int @ Q4 @ L2 ) ) )
          | ? [R5: int,L2: int,K2: int,Q4: int] :
              ( ( A12 = K2 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K2
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9337_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A33 )
     => ( ( ( A22 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q2: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q2 @ A22 ) ) ) )
         => ~ ! [R3: int,Q2: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q2 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9338_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9339_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K2: int] :
          ( plus_plus_int @ K2
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% set_bit_eq
thf(fact_9340_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9341_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X2 )
          & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X2 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9342_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X2 )
            & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X2 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9343_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9344_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9345_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_9346_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9347_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9348_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9349_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).

% real_sgn_eq
thf(fact_9350_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9351_sgn__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9352_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9353_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9354_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M5: nat,N3: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9355_sgn__power__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9356_root__sgn__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9357_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_9358_split__root,axiom,
    ! [P: real > $o,N2: nat,X: real] :
      ( ( P @ ( root @ N2 @ X ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
                = X )
             => ( P @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_9359_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X2: real] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X2 )
              & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9360_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9361_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_9362_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K2: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K2
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9363_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K2: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9364_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9365_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9366_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9367_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_9368_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9369_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_9370_diff__nat__numeral,axiom,
    ! [V: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_9371_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9372_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9373_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9374_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_9375_nat__numeral__diff__1,axiom,
    ! [V: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_9376_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9377_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9378_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9379_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9380_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9381_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_9382_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9383_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M5 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9384_nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% nat_mask_eq
thf(fact_9385_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_9386_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_9387_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_9388_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_9389_int__minus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_9390_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9391_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9392_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% and_nat_def
thf(fact_9393_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9394_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% or_nat_def
thf(fact_9395_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9396_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9397_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9398_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_9399_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_9400_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_9401_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_9402_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9403_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9404_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9405_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9406_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9407_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9408_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_9409_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_9410_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_9411_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9412_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9413_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_9414_floor__eq3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_9415_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9416_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_9417_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9418_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9419_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_9420_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9421_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_9422_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_9423_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9424_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9425_floor__eq4,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_9426_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9427_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_9428_powr__real__of__int,axiom,
    ! [X: real,N2: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
            = ( power_power_real @ X @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9429_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X2: rat] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 )
              & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9430_Arg__def,axiom,
    ( arg
    = ( ^ [Z3: complex] :
          ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A3: real] :
                ( ( ( sgn_sgn_complex @ Z3 )
                  = ( cis @ A3 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
                & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9431_cis__multiple__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9432_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ S @ M )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
           => ( vEBT_invar_vebt @ X3 @ N2 ) )
       => ( ( M
            = ( suc @ N2 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
               => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X3 ) )
                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
           => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
             => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9433_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% height_compose_list
thf(fact_9434_max__ins__scaled,axiom,
    ! [N2: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_9435_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).

% height_i_max
thf(fact_9436_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N2: nat,X14: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N2 @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).

% max_idx_list
thf(fact_9437_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_9438_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S2 )
           => ! [T3: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T3 )
               => ( R2
                 != ( plus_plus_rat @ S2 @ T3 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9439_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_9440_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( ? [A5: $o,B6: $o] :
              ( X
              = ( vEBT_Leaf @ A5 @ B6 ) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_9441_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9442_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N3 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9443_sin__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9444_cos__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9445_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_9446_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9447_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ N2 ) ) )
        = ( abs_abs_int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_9448_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9449_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_9450_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_9451_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9452_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9453_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9454_rat__inverse__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_inverse_code
thf(fact_9455_normalize__negative,axiom,
    ! [Q3: int,P2: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_9456_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9457_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9458_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_9459_normalize__denom__zero,axiom,
    ! [P2: int] :
      ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9460_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9461_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9462_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9463_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( times_times_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B2 ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_times_code
thf(fact_9464_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_divide_code
thf(fact_9465_quotient__of__div,axiom,
    ! [R2: rat,N2: int,D: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N2 @ D ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9466_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_minus_code
thf(fact_9467_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q3: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% quotient_of_denom_pos
thf(fact_9468_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_plus_code
thf(fact_9469_rat__uminus__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_uminus_code
thf(fact_9470_rat__abs__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_abs_code
thf(fact_9471_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q3: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_9472_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P2 @ S )
            = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9473_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9474_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P5: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P5 ) ) ) ) ).

% rat_floor_code
thf(fact_9475_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9476_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9477_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9478_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9479_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9480_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9481_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9482_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9483_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9484_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9485_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_9486_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_9487_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9488_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9489_concat__bit__of__zero__1,axiom,
    ! [N2: nat,L: int] :
      ( ( bit_concat_bit @ N2 @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N2 @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_9490_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9491_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9492_push__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_Suc_0
thf(fact_9493_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
       != ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).

% bit_xor_int_iff
thf(fact_9494_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9495_push__bit__nat__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9496_XOR__lower,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 @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9497_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M5: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9498_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M5: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9499_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9500_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9501_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9502_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N3: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_9503_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N3: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K2 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).

% concat_bit_def
thf(fact_9504_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9505_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_int_def
thf(fact_9506_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9507_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9508_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9509_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9510_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9511_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_9512_push__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% push_bit_minus_one
thf(fact_9513_XOR__upper,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_9514_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_9515_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9516_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9517_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( K2
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K2 )
            @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9518_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_9519_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9520_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9521_or__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9522_and__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9523_bit__not__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N2 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_not_int_iff
thf(fact_9524_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).

% or_int_def
thf(fact_9525_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9526_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_9527_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_9528_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9529_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).

% xor_int_def
thf(fact_9530_not__int__div__2,axiom,
    ! [K: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_9531_even__not__iff__int,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_not_iff_int
thf(fact_9532_and__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(4)
thf(fact_9533_and__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9534_or__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9535_or__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9536_bit__minus__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N2 )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N2 ) ) ).

% bit_minus_int_iff
thf(fact_9537_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9538_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9539_numeral__or__not__num__eq,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9540_and__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9541_and__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(7)
thf(fact_9542_or__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9543_and__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9544_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9545_and__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9546_and__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9547_or__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9548_or__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9549_and__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9550_or__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9551_or__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9552_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9553_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( ( B6
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B6
                   => ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9554_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( ( A5
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A5
                   => ( ( B6
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B6
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9555_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A5 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_9556_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
       => ( ! [A5: $o,B6: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B6 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B6 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A5 @ B6 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_9557_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_9558_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9559_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9560_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9561_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X4: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M8 @ M5 )
         => ! [N3: nat] :
              ( ( ord_less_eq_nat @ M8 @ N3 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X4 @ M5 ) @ ( X4 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9562_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9563_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9564_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9565_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N2 )
            & ( P @ M5 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9566_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N2 )
           => ( P @ M5 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9567_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9568_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9569_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9570_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9571_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_Suc_fact
thf(fact_9572_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9573_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9574_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9575_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9576_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9577_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9578_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X2: int] : ( plus_plus_int @ X2 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9579_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size(3)
thf(fact_9580_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size_gen(1)
thf(fact_9581_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9582_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9583_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9584_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9585_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_9586_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
thf(fact_9587_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] :
      ( ( re @ ( csqrt @ Z ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_9588_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9589_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_9590_Re__divide__of__nat,axiom,
    ! [Z: complex,N2: nat] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% Re_divide_of_nat
thf(fact_9591_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).

% Re_divide_of_real
thf(fact_9592_Re__sgn,axiom,
    ! [Z: complex] :
      ( ( re @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Re_sgn
thf(fact_9593_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Re_divide_numeral
thf(fact_9594_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9595_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9596_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9597_sums__Re,axiom,
    ! [X8: nat > complex,A: complex] :
      ( ( sums_complex @ X8 @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( re @ ( X8 @ N3 ) )
        @ ( re @ A ) ) ) ).

% sums_Re
thf(fact_9598_Cauchy__Re,axiom,
    ! [X8: nat > complex] :
      ( ( topolo6517432010174082258omplex @ X8 )
     => ( topolo4055970368930404560y_real
        @ ^ [N3: nat] : ( re @ ( X8 @ N3 ) ) ) ) ).

% Cauchy_Re
thf(fact_9599_complex__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_Re_le_cmod
thf(fact_9600_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_9601_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% plus_complex.simps(1)
thf(fact_9602_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
      = ( times_times_real @ R2 @ ( re @ X ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9603_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% minus_complex.simps(1)
thf(fact_9604_summable__Re,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_real
        @ ^ [X2: nat] : ( re @ ( F @ X2 ) ) ) ) ).

% summable_Re
thf(fact_9605_abs__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Re_le_cmod
thf(fact_9606_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9607_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9608_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_9609_cos__n__Re__cis__pow__n,axiom,
    ! [N2: nat,A: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) )
      = ( re @ ( power_power_complex @ ( cis @ A ) @ N2 ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_9610_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z3: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z3 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9611_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      ( ( im @ ( csqrt @ Z ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_9612_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K2: int] :
          ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
          @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9613_Im__i__times,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( re @ Z ) ) ).

% Im_i_times
thf(fact_9614_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).

% Im_divide_of_real
thf(fact_9615_Im__sgn,axiom,
    ! [Z: complex] :
      ( ( im @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Im_sgn
thf(fact_9616_Re__power__real,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N2 ) )
        = ( power_power_real @ ( re @ X ) @ N2 ) ) ) ).

% Re_power_real
thf(fact_9617_Re__i__times,axiom,
    ! [Z: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( uminus_uminus_real @ ( im @ Z ) ) ) ).

% Re_i_times
thf(fact_9618_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Im_divide_numeral
thf(fact_9619_Im__divide__of__nat,axiom,
    ! [Z: complex,N2: nat] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% Im_divide_of_nat
thf(fact_9620_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
       => ( ( csqrt @ X )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9621_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_9622_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9623_divide__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa2 @ X ) ) ) ).

% divide_integer.abs_eq
thf(fact_9624_sums__Im,axiom,
    ! [X8: nat > complex,A: complex] :
      ( ( sums_complex @ X8 @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( im @ ( X8 @ N3 ) )
        @ ( im @ A ) ) ) ).

% sums_Im
thf(fact_9625_Cauchy__Im,axiom,
    ! [X8: nat > complex] :
      ( ( topolo6517432010174082258omplex @ X8 )
     => ( topolo4055970368930404560y_real
        @ ^ [N3: nat] : ( im @ ( X8 @ N3 ) ) ) ) ).

% Cauchy_Im
thf(fact_9626_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_9627_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_9628_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% plus_complex.simps(2)
thf(fact_9629_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
      = ( times_times_real @ R2 @ ( im @ X ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9630_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% minus_complex.simps(2)
thf(fact_9631_sums__complex__iff,axiom,
    ( sums_complex
    = ( ^ [F3: nat > complex,X2: complex] :
          ( ( sums_real
            @ ^ [Y2: nat] : ( re @ ( F3 @ Y2 ) )
            @ ( re @ X2 ) )
          & ( sums_real
            @ ^ [Y2: nat] : ( im @ ( F3 @ Y2 ) )
            @ ( im @ X2 ) ) ) ) ) ).

% sums_complex_iff
thf(fact_9632_summable__Im,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_real
        @ ^ [X2: nat] : ( im @ ( F @ X2 ) ) ) ) ).

% summable_Im
thf(fact_9633_abs__Im__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Im_le_cmod
thf(fact_9634_summable__complex__iff,axiom,
    ( summable_complex
    = ( ^ [F3: nat > complex] :
          ( ( summable_real
            @ ^ [X2: nat] : ( re @ ( F3 @ X2 ) ) )
          & ( summable_real
            @ ^ [X2: nat] : ( im @ ( F3 @ X2 ) ) ) ) ) ) ).

% summable_complex_iff
thf(fact_9635_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( times_times_complex @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9636_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( im @ X )
        = ( im @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_9637_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( re @ X )
        = ( re @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_9638_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( times_times_complex @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9639_plus__complex_Ocode,axiom,
    ( plus_plus_complex
    = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).

% plus_complex.code
thf(fact_9640_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9641_minus__complex_Ocode,axiom,
    ( minus_minus_complex
    = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).

% minus_complex.code
thf(fact_9642_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_9643_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_9644_sin__n__Im__cis__pow__n,axiom,
    ! [N2: nat,A: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) )
      = ( im @ ( power_power_complex @ ( cis @ A ) @ N2 ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_9645_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_9646_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_9647_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_9648_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9649_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z3: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z3 ) ) ) @ ( cis @ ( im @ Z3 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_9650_cmod__power2,axiom,
    ! [Z: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_9651_Im__power2,axiom,
    ! [X: complex] :
      ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).

% Im_power2
thf(fact_9652_Re__power2,axiom,
    ! [X: complex] :
      ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9653_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_9654_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z3: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9655_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X ) )
      = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9656_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_9657_Re__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_9658_csqrt__square,axiom,
    ! [B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
        | ( ( ( re @ B )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B ) ) ).

% csqrt_square
thf(fact_9659_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z )
          = W ) ) ) ).

% csqrt_unique
thf(fact_9660_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(2)
thf(fact_9661_Im__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_9662_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_9663_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_9664_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9665_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9666_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9667_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9668_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
        = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).

% Re_divide_Reals
thf(fact_9669_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9670_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9671_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
        = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).

% Im_divide_Reals
thf(fact_9672_complex__diff__cnj,axiom,
    ! [Z: complex] :
      ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_9673_complex__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_9674_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( times_times_complex @ X @ Y ) )
      = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_mult
thf(fact_9675_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_9676_sums__cnj,axiom,
    ! [F: nat > complex,L: complex] :
      ( ( sums_complex
        @ ^ [X2: nat] : ( cnj @ ( F @ X2 ) )
        @ ( cnj @ L ) )
      = ( sums_complex @ F @ L ) ) ).

% sums_cnj
thf(fact_9677_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_9678_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_9679_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z3: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_9680_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_9681_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_9682_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_9683_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_9684_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_9685_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_9686_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_9687_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_9688_complex__mod__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_9689_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_9690_complex__norm__square,axiom,
    ! [Z: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).

% complex_norm_square
thf(fact_9691_complex__add__cnj,axiom,
    ! [Z: complex] :
      ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).

% complex_add_cnj
thf(fact_9692_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9693_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_9694_integer__of__num_I3_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_9695_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K2: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9696_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K2: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9697_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9698_integer__of__num_I2_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).

% integer_of_num(2)
thf(fact_9699_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9700_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L2 ) @ ( modulo364778990260209775nteger @ K2 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_9701_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K2: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9702_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K2: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9703_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K2: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
          @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9704_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_9705_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_9706_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa2 ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% divide_integer.rep_eq
thf(fact_9707_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_of_integer_code_post(3)
thf(fact_9708_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_9709_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_9710_nat__of__integer__code__post_I2_J,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_code_post(2)
thf(fact_9711_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9712_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
                @ ( code_divmod_abs @ K2 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
                    @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9713_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_9714_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9715_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_9716_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_9717_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).

% card_atLeastAtMost
thf(fact_9718_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu3: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_9719_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu3: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_9720_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9721_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M7 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9722_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ ( suc @ K2 ) @ M7 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9723_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
     => ( A2
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9724_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_9725_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N2 @ M6 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9726_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9727_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9728_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9729_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_9730_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z3: complex] :
                  ( ( power_power_complex @ Z3 @ N2 )
                  = C ) ) )
          = N2 ) ) ) ).

% card_nth_roots
thf(fact_9731_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = one_one_complex ) ) )
        = N2 ) ) ).

% card_roots_unity_eq
thf(fact_9732_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K2: nat] : K2
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9733_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9734_binomial__def,axiom,
    ( binomial
    = ( ^ [N3: nat,K2: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K2 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9735_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9736_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9737_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9738_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9739_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9740_drop__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9741_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9742_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9743_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9744_drop__bit__push__bit__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N2 @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9745_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N3: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9746_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K2 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K2 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M5 @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9747_Suc__0__mod__numeral,axiom,
    ! [K: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9748_Suc__0__div__numeral,axiom,
    ! [K: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_div_numeral
thf(fact_9749_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9750_fst__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( divide_divide_nat @ M @ N2 ) ) ).

% fst_divmod_nat
thf(fact_9751_snd__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ M @ N2 ) ) ).

% snd_divmod_nat
thf(fact_9752_drop__bit__nat__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N2 @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9753_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N3: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9754_rat__sgn__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P2 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P2 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9755_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9756_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X2: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9757_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9758_one__mod__minus__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9759_minus__one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9760_numeral__mod__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9761_minus__numeral__mod__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9762_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9763_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9764_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9765_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N2: num] :
      ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N2 ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_9766_gcd__neg__numeral__1__int,axiom,
    ! [N2: num,X: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X ) ) ).

% gcd_neg_numeral_1_int
thf(fact_9767_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9768_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N2 ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9769_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N2 )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] :
                ( ( dvd_dvd_int @ D2 @ M )
                & ( dvd_dvd_int @ D2 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_9770_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N2 @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K3 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9771_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9772_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_9773_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9774_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9775_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9776_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9777_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_9778_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9779_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_9780_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_9781_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9782_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9783_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M )
                & ( dvd_dvd_nat @ D2 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9784_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9785_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M5 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9786_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M3 @ N9 )
              & ( member_nat @ N9 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_9787_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9788_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S3 ) )
             => ( member_nat @ ( R3 @ N9 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_9789_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N2: num] :
      ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9790_xor__minus__numerals_I1_J,axiom,
    ! [N2: num,K: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ K )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N2 @ one ) @ K ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9791_sub__BitM__One__eq,axiom,
    ! [N2: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9792_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9793_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
    @ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9794_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M5: nat,N3: nat] :
        ( ( dvd_dvd_nat @ M5 @ N3 )
        & ( M5 != N3 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9795_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K2 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K2 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
                  @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9796_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9797_card_Ocomp__fun__commute__on,axiom,
    ( ( comp_nat_nat_nat @ suc @ suc )
    = ( comp_nat_nat_nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_9798_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y3: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_9799_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_9800_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9801_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9802_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_9803_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9804_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9805_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9806_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9807_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9808_Code__Target__Int_Onegative__def,axiom,
    ( code_Target_negative
    = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).

% Code_Target_Int.negative_def
thf(fact_9809_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9810_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
      = Q3 ) ).

% num_of_nat_numeral_eq
thf(fact_9811_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9812_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_9813_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_9814_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_9815_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9816_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z3: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9817_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z3: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9818_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M5: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N3 ) ) @ M5 ) ) ) ).

% prod_encode_def
thf(fact_9819_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_9820_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_9821_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).

% nat.rep_eq
thf(fact_9822_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_9823_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) ) ) ) ).

% uminus_int_def
thf(fact_9824_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9825_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9826_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9827_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_9828_sqr_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N2 ) ) ) ) ).

% sqr.simps(2)
thf(fact_9829_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9830_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).

% sqr_conv_mult
thf(fact_9831_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9832_sqr_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).

% sqr.simps(3)
thf(fact_9833_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
      & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9834_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).

% mult_rat
thf(fact_9835_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).

% divide_rat
thf(fact_9836_floor__Fract,axiom,
    ! [A: int,B: int] :
      ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
      = ( divide_divide_int @ A @ B ) ) ).

% floor_Fract
thf(fact_9837_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9838_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% add_rat
thf(fact_9839_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9840_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9841_sgn__rat,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).

% sgn_rat
thf(fact_9842_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9843_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( fract @ A @ B ) ) ) ).

% mult_rat_cancel
thf(fact_9844_Fract__coprime,axiom,
    ! [A: int,B: int] :
      ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
      = ( fract @ A @ B ) ) ).

% Fract_coprime
thf(fact_9845_quotient__of__eq,axiom,
    ! [A: int,B: int,P2: int,Q3: int] :
      ( ( ( quotient_of @ ( fract @ A @ B ) )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ( fract @ P2 @ Q3 )
        = ( fract @ A @ B ) ) ) ).

% quotient_of_eq
thf(fact_9846_normalize__eq,axiom,
    ! [A: int,B: int,P2: int,Q3: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ( fract @ P2 @ Q3 )
        = ( fract @ A @ B ) ) ) ).

% normalize_eq
thf(fact_9847_rat__number__collapse_I3_J,axiom,
    ! [W: num] :
      ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
      = ( numeral_numeral_rat @ W ) ) ).

% rat_number_collapse(3)
thf(fact_9848_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_9849_quotient__of__Fract,axiom,
    ! [A: int,B: int] :
      ( ( quotient_of @ ( fract @ A @ B ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).

% quotient_of_Fract
thf(fact_9850_rat__number__collapse_I4_J,axiom,
    ! [W: num] :
      ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% rat_number_collapse(4)
thf(fact_9851_rat__number__expand_I5_J,axiom,
    ! [K: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
      = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% rat_number_expand(5)
thf(fact_9852_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X4: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X4 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9853_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_9854_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_9855_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_9856_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9857_suminf__eq__SUP__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I2 ) )
       => ( ( suminf_real @ X8 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9858_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9859_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9860_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9861_root__def,axiom,
    ( root
    = ( ^ [N3: nat,X2: real] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N3 ) )
            @ X2 ) ) ) ) ).

% root_def
thf(fact_9862_card__UNIV__char,axiom,
    ( ( finite_card_char @ top_top_set_char )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% card_UNIV_char
thf(fact_9863_UNIV__char__of__nat,axiom,
    ( top_top_set_char
    = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_9864_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_9865_range__nat__of__char,axiom,
    ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
    = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_9866_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_9867_char__of__integer__code,axiom,
    ( char_of_integer
    = ( ^ [K2: code_integer] :
          ( produc4188289175737317920o_char
          @ ^ [Q0: code_integer,B02: $o] :
              ( produc4188289175737317920o_char
              @ ^ [Q1: code_integer,B12: $o] :
                  ( produc4188289175737317920o_char
                  @ ^ [Q22: code_integer,B23: $o] :
                      ( produc4188289175737317920o_char
                      @ ^ [Q32: code_integer,B33: $o] :
                          ( produc4188289175737317920o_char
                          @ ^ [Q42: code_integer,B43: $o] :
                              ( produc4188289175737317920o_char
                              @ ^ [Q52: code_integer,B53: $o] :
                                  ( produc4188289175737317920o_char
                                  @ ^ [Q62: code_integer,B63: $o] :
                                      ( produc4188289175737317920o_char
                                      @ ^ [Uu3: code_integer] : ( char2 @ B02 @ B12 @ B23 @ B33 @ B43 @ B53 @ B63 )
                                      @ ( code_bit_cut_integer @ Q62 ) )
                                  @ ( code_bit_cut_integer @ Q52 ) )
                              @ ( code_bit_cut_integer @ Q42 ) )
                          @ ( code_bit_cut_integer @ Q32 ) )
                      @ ( code_bit_cut_integer @ Q22 ) )
                  @ ( code_bit_cut_integer @ Q1 ) )
              @ ( code_bit_cut_integer @ Q0 ) )
          @ ( code_bit_cut_integer @ K2 ) ) ) ) ).

% char_of_integer_code
thf(fact_9868_String_Ochar__of__ascii__of,axiom,
    ! [C: char] :
      ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
      = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).

% String.char_of_ascii_of
thf(fact_9869_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_9870_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_9871_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9872_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs3: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_9873_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9874_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9875_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9876_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9877_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_9878_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9879_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9880_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9881_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9882_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9883_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9884_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_9885_upto__code,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_9886_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9887_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_9888_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9889_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9890_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9891_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa2 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa2 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9892_upto_Osimps,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9893_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9894_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9895_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9896_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_9897_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9898_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M5: nat] :
                      ( collect_nat
                      @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M5 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9899_Gcd__nat__eq__one,axiom,
    ! [N5: set_nat] :
      ( ( member_nat @ one_one_nat @ N5 )
     => ( ( gcd_Gcd_nat @ N5 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_9900_DERIV__real__root__generic,axiom,
    ! [N2: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9901_Gcd__int__eq,axiom,
    ! [N5: set_nat] :
      ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N5 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N5 ) ) ) ).

% Gcd_int_eq
thf(fact_9902_Gcd__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
      = ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_abs_eq
thf(fact_9903_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_nat
        @ ( image_int_nat
          @ ^ [K2: int] : ( nat2 @ ( abs_abs_int @ K2 ) )
          @ K5 ) )
      = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).

% Gcd_nat_abs_eq
thf(fact_9904_DERIV__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9905_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9906_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9907_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9908_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9909_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_9910_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9911_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9912_DERIV__mirror,axiom,
    ! [F: real > real,Y: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
      = ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
        @ ( uminus_uminus_real @ Y )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_mirror
thf(fact_9913_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9914_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9915_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9916_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9917_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9918_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z2: real] :
            ( ( ord_less_real @ A @ Z2 )
            & ( ord_less_real @ Z2 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).

% MVT2
thf(fact_9919_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9920_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9921_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9922_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9923_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9924_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9925_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9926_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9927_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9928_DERIV__pow,axiom,
    ! [N2: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X2: real] : ( power_power_real @ X2 @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_9929_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N2: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N2 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9930_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9931_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9932_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9933_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9934_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9935_DERIV__arctan,axiom,
    ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9936_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9937_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9938_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9939_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9940_Gcd__int__def,axiom,
    ( gcd_Gcd_int
    = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).

% Gcd_int_def
thf(fact_9941_DERIV__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9942_DERIV__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9943_DERIV__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9944_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9945_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9946_DERIV__odd__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9947_Maclaurin,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T3: real] :
                ( ( ( ord_less_nat @ M3 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
                  & ( ord_less_eq_real @ T3 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ T3 )
                & ( ord_less_real @ T3 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9948_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T3: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T3 )
                & ( ord_less_eq_real @ T3 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ? [T3: real] :
              ( ( ord_less_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9949_Maclaurin__minus,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T3: real] :
                ( ( ( ord_less_nat @ M3 @ N2 )
                  & ( ord_less_eq_real @ H2 @ T3 )
                  & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ H2 @ T3 )
                & ( ord_less_real @ T3 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9950_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
                & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9951_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,T3: real] :
            ( ( ( ord_less_nat @ M3 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9952_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T3: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T3: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T3 )
                            & ( ord_less_real @ T3 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T3 )
                            & ( ord_less_real @ T3 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M5 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9953_Taylor__up,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T3: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T3: real] :
                  ( ( ord_less_real @ C @ T3 )
                  & ( ord_less_real @ T3 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9954_Taylor__down,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T3: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T3: real] :
                  ( ( ord_less_real @ A @ T3 )
                  & ( ord_less_real @ T3 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9955_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
      ( ! [M3: nat,T3: real] :
          ( ( ( ord_less_nat @ M3 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M4: nat,T4: real] :
            ( ( ( ord_less_nat @ M4 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M4 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M4 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M4 ) ) )
                    @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M4 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M4 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M4 ) @ T4 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M4 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T4 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M4 ) ) ) )
                  @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N2 @ ( suc @ M4 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M4 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9956_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9957_DERIV__even__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9958_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X3 @ N3 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] :
                ( suminf_real
                @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X2 @ ( suc @ N3 ) ) ) )
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9959_tanh__real__bounds,axiom,
    ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9960_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9961_DERIV__series_H,axiom,
    ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( F @ X2 @ N )
          @ ( F4 @ X0 @ N )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F4 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N: nat,X3: real,Y3: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N ) @ ( F @ Y3 @ N ) ) ) @ ( times_times_real @ ( L5 @ N ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
                  @ ( suminf_real @ ( F4 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9962_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9963_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9964_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9965_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9966_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9967_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).

% card_greaterThanLessThan
thf(fact_9968_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9969_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9970_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9971_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9972_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9973_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9974_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9975_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N3 @ M ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9976_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9977_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9978_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_9979_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9980_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ N2 @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ one )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9981_and__not__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9982_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9983_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_9984_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q3: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( some_num @ Q3 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = ( numeral_numeral_int @ Q3 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9985_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_9986_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9987_and__not__num__eq__None__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9988_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9989_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9990_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_9991_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M5: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M5 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M5 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9992_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9993_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M5: num] :
          ( produc478579273971653890on_num
          @ ^ [A3: nat,X2: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P5: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X2 )
              @ A3 )
          @ ( product_Pair_nat_num @ N3 @ M5 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9994_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: real] :
            ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A @ X5 )
                  & ( ord_less_eq_real @ X5 @ B ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
                  & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ M9 ) )
               => ? [X3: real] :
                    ( ( ord_less_eq_real @ A @ X3 )
                    & ( ord_less_eq_real @ X3 @ B )
                    & ( ( F @ X3 )
                      = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9995_isCont__real__sqrt,axiom,
    ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9996_isCont__real__root,axiom,
    ! [X: real,N2: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N2 ) ) ).

% isCont_real_root
thf(fact_9997_continuous__frac,axiom,
    ! [X: real] :
      ( ~ ( member_real @ X @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).

% continuous_frac
thf(fact_9998_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B )
               => ( ( G @ ( F @ Z2 ) )
                  = Z2 ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_eq_real @ A @ Z2 )
               => ( ( ord_less_eq_real @ Z2 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9999_isCont__arcosh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_10000_continuous__floor,axiom,
    ! [X: real] :
      ( ~ ( member_real @ X @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).

% continuous_floor
thf(fact_10001_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X )
         => ( ( ord_less_real @ X @ B )
           => ( ! [Y3: real] :
                  ( ( ord_less_real @ A @ Y3 )
                 => ( ( ord_less_real @ Y3 @ B )
                   => ( ( F @ ( G @ Y3 ) )
                      = Y3 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_10002_isCont__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_10003_isCont__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_10004_LIM__less__bound,axiom,
    ! [B: real,X: real,F: real > real] :
      ( ( ord_less_real @ B @ X )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_10005_isCont__artanh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_10006_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
           => ( ( G @ ( F @ Z2 ) )
              = Z2 ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_10007_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ A @ Z2 )
           => ( ( ord_less_eq_real @ Z2 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
               => ( ( ord_less_real @ Z2 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
           => ( ! [Z2: real] :
                  ( ( ord_less_real @ A @ Z2 )
                 => ( ( ord_less_real @ Z2 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_10008_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_10009_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ( F @ X5 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_10010_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_10011_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_10012_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_10013_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_10014_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_10015_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_10016_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_10017_monoseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( topolo6980174941875973593q_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I2 ) ) @ B3 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_10018_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_10019_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N ) ) @ ( G @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
         => ( ( filterlim_nat_real
              @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_10020_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N7 @ N )
         => ( ord_less_real @ R3 @ ( X8 @ N ) ) )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( X8 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_10021_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_10022_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( root @ N3 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_10023_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_10024_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_10025_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ L )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_10026_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_10027_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_10028_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_10029_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_10030_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10031_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10032_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_10033_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10034_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10035_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).

% summable
thf(fact_10036_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K3: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10037_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K3: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10038_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10039_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10040_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10041_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10042_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N9: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N9: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10043_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10044_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10045_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_10046_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F3: real > real] :
        ? [C3: real] :
          ( F3
          = ( ^ [X2: real] : ( times_times_real @ X2 @ C3 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10047_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ one_one_real @ Y2 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10048_dist__real__def,axiom,
    ( real_V975177566351809787t_real
    = ( ^ [X2: real,Y2: real] : ( abs_abs_real @ ( minus_minus_real @ X2 @ Y2 ) ) ) ) ).

% dist_real_def
thf(fact_10049_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).

% tendsto_arcosh_at_left_1
thf(fact_10050_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10051_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_10052_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_10053_filterlim__tan__at__right,axiom,
    filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% filterlim_tan_at_right
thf(fact_10054_tendsto__arctan__at__bot,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).

% tendsto_arctan_at_bot
thf(fact_10055_tanh__real__at__bot,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).

% tanh_real_at_bot
thf(fact_10056_ln__at__0,axiom,
    filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% ln_at_0
thf(fact_10057_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_bot_neg
thf(fact_10058_artanh__real__at__right__1,axiom,
    filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% artanh_real_at_right_1
thf(fact_10059_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10060_filterlim__pow__at__bot__odd,axiom,
    ! [N2: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
            @ at_bot_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10061_filterlim__pow__at__bot__even,axiom,
    ! [N2: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
            @ at_top_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10062_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_10063_ln__at__top,axiom,
    filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).

% ln_at_top
thf(fact_10064_exp__at__top,axiom,
    filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).

% exp_at_top
thf(fact_10065_filterlim__real__sequentially,axiom,
    filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).

% filterlim_real_sequentially
thf(fact_10066_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).

% filterlim_uminus_at_bot_at_top
thf(fact_10067_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).

% filterlim_uminus_at_top_at_bot
thf(fact_10068_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_10069_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_10070_artanh__real__at__left__1,axiom,
    filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).

% artanh_real_at_left_1
thf(fact_10071_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).

% filterlim_inverse_at_right_top
thf(fact_10072_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_top_right
thf(fact_10073_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10074_tendsto__exp__limit__at__top,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y2 ) ) @ Y2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10075_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B @ X3 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_10076_tendsto__arctan__at__top,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).

% tendsto_arctan_at_top
thf(fact_10077_filterlim__tan__at__left,axiom,
    filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% filterlim_tan_at_left
thf(fact_10078_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10079_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10080_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10081_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10082_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N3: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_10083_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C @ X3 )
         => ( P @ X3 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10084_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N6: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( P @ N3 ) ) ) ) ).

% eventually_sequentially
thf(fact_10085_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
      = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).

% le_sequentially
thf(fact_10086_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_10087_eventually__at__left__to__right,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( uminus_uminus_real @ X2 ) )
        @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).

% eventually_at_left_to_right
thf(fact_10088_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( eventually_real
        @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_10089_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( eventually_real
        @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_10090_eventually__at__right__to__top,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_10091_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_10092_lhopital__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_10093_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10094_lhopital__right__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_10095_lhopital__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_10096_lhopital__left__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_10097_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10098_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10099_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_10100_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G0 @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_10101_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10102_lhopital__right__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_10103_lhopital__left__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_10104_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_10105_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_10106_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10107_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_10108_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_10109_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_10110_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_10111_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_10112_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_10113_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_10114_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10115_decseq__bounded,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I2 ) )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10116_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_10117_decseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I2 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I4: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I4 ) ) ) ) ) ).

% decseq_convergent
thf(fact_10118_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_10119_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_10120_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
                          & ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X4 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X2: nat] :
                                      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                       => ( ( ord_less_nat @ Mi3 @ X2 )
                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_10121_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X4 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X2: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X2: nat] :
                                  ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                   => ( ( ord_less_nat @ Mi3 @ X2 )
                                      & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_10122_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X4 )
            & ! [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X4 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                    & ! [X2: nat] :
                        ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X2 )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_10123_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
                    & ! [X2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I3: nat] :
                            ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X4 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X2: nat] :
                                ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                 => ( ( ord_less_nat @ Mi3 @ X2 )
                                    & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_10124_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
                          & ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X4 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X2: nat] :
                                      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                       => ( ( ord_less_nat @ Mi3 @ X2 )
                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_10125_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
                          & ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X4 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X2: nat] :
                                      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                       => ( ( ord_less_nat @ Mi3 @ X2 )
                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_10126_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X4: set_int] :
          ( the_int
          @ ^ [X2: int] :
              ( ( member_int @ X2 @ X4 )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ X4 )
                 => ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_10127_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X4 )
                        & ! [X2: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I3: nat] :
                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X4 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X4 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X2: nat] :
                                    ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                     => ( ( ord_less_nat @ Mi3 @ X2 )
                                        & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_10128_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ( ! [X3: real] :
              ( ( ( ord_less_real @ A @ X3 )
                & ( ord_less_real @ X3 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
         => ( ! [X3: real] :
                ( ( ( ord_less_eq_real @ A @ X3 )
                  & ( ord_less_eq_real @ X3 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
           => ( ! [X3: real] :
                  ( ( ( ord_less_real @ A @ X3 )
                    & ( ord_less_real @ X3 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10129_real__differentiable__def,axiom,
    ! [F: real > real,X: real,S: set_real] :
      ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
      = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ) ).

% real_differentiable_def
thf(fact_10130_real__differentiableE,axiom,
    ! [F: real > real,X: real,S: set_real] :
      ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ~ ! [Df: real] :
            ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).

% real_differentiableE
thf(fact_10131_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z2: real] :
              ( ( ord_less_real @ A @ Z2 )
              & ( ord_less_real @ Z2 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10132_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_10133_continuous__on__arsinh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( topolo5044208981011980120l_real @ A2
        @ ^ [X2: real] : ( arsinh_real @ ( F @ X2 ) ) ) ) ).

% continuous_on_arsinh'
thf(fact_10134_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10135_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ? [C2: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
              = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
            & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10136_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).

% continuous_on_arccos'
thf(fact_10137_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).

% continuous_on_arcsin'
thf(fact_10138_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_10139_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_10140_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B )
                & ( ( F4 @ Z2 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10141_mvt,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10142_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10143_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_10144_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( F @ B )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10145_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X )
           => ( ( ord_less_eq_real @ X @ B )
             => ( ( F @ X )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10146_Rolle,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10147_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X2: complex,Y2: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X2 @ Y2 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_10148_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X2: real,Y2: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X2 @ Y2 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_10149_open__real__def,axiom,
    ( topolo4860482606490270245n_real
    = ( ^ [U4: set_real] :
        ! [X2: real] :
          ( ( member_real @ X2 @ U4 )
         => ( eventu3244425730907250241l_real
            @ ( produc5414030515140494994real_o
              @ ^ [X9: real,Y2: real] :
                  ( ( X9 = X2 )
                 => ( member_real @ Y2 @ U4 ) ) )
            @ topolo1511823702728130853y_real ) ) ) ) ).

% open_real_def
thf(fact_10150_open__complex__def,axiom,
    ( topolo4110288021797289639omplex
    = ( ^ [U4: set_complex] :
        ! [X2: complex] :
          ( ( member_complex @ X2 @ U4 )
         => ( eventu5826381225784669381omplex
            @ ( produc6771430404735790350plex_o
              @ ^ [X9: complex,Y2: complex] :
                  ( ( X9 = X2 )
                 => ( member_complex @ Y2 @ U4 ) ) )
            @ topolo896644834953643431omplex ) ) ) ) ).

% open_complex_def
thf(fact_10151_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N6: nat] :
          ! [M5: nat] :
            ( ( ord_less_eq_nat @ N6 @ M5 )
           => ! [N3: nat] :
                ( ( ord_less_eq_nat @ N6 @ N3 )
               => ( P @ ( product_Pair_nat_nat @ N3 @ M5 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10152_incseq__bounded,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B3 )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_10153_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10154_mono__times__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).

% mono_times_nat
thf(fact_10155_filtermap__at__right__shift,axiom,
    ! [D: real,A: real] :
      ( ( filtermap_real_real
        @ ^ [X2: real] : ( minus_minus_real @ X2 @ D )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).

% filtermap_at_right_shift
thf(fact_10156_incseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ B3 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_10157_at__right__to__0,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
      = ( filtermap_real_real
        @ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_right_to_0
thf(fact_10158_at__right__minus,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
      = ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5984915006950818249n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).

% at_right_minus
thf(fact_10159_at__left__minus,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) )
      = ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).

% at_left_minus
thf(fact_10160_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M5: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M5 ) @ M5 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10161_tendsto__at__topI__sequentially__real,axiom,
    ! [F: real > real,Y: real] :
      ( ( order_mono_real_real @ F )
     => ( ( filterlim_nat_real
          @ ^ [N3: nat] : ( F @ ( semiri5074537144036343181t_real @ N3 ) )
          @ ( topolo2815343760600316023s_real @ Y )
          @ at_top_nat )
       => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).

% tendsto_at_topI_sequentially_real
thf(fact_10162_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] :
                  ( Xa2
                  = ( bit0 @ N ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] :
                    ( Xa2
                    = ( bit1 @ N ) )
               => ( Y != none_num ) ) )
           => ( ! [M3: num] :
                  ( ( X
                    = ( bit0 @ M3 ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M3 ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ! [N: num] :
                        ( ( Xa2
                          = ( bit0 @ N ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N: num] :
                          ( ( Xa2
                            = ( bit1 @ N ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit1 @ M3 ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M3 ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ! [N: num] :
                              ( ( Xa2
                                = ( bit0 @ N ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M3 @ N ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N: num] :
                                ( ( Xa2
                                  = ( bit1 @ N ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_10163_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(5)
thf(fact_10164_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(6)
thf(fact_10165_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(9)
thf(fact_10166_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] :
                  ( Xa2
                  = ( bit0 @ N ) )
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] :
                    ( Xa2
                    = ( bit1 @ N ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M3: num] :
                    ( X
                    = ( bit0 @ M3 ) )
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ! [N: num] :
                        ( ( Xa2
                          = ( bit0 @ N ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N: num] :
                          ( ( Xa2
                            = ( bit1 @ N ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N ) ) ) ) )
                 => ( ( ? [M3: num] :
                          ( X
                          = ( bit1 @ M3 ) )
                     => ( ( Xa2 = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ! [N: num] :
                              ( ( Xa2
                                = ( bit0 @ N ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N: num] :
                                ( ( Xa2
                                  = ( bit1 @ N ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M3 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_10167_and__num_Osimps_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(5)
thf(fact_10168_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_10169_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_10170_and__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_10171_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_10172_and__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_10173_and__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(6)
thf(fact_10174_and__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(8)
thf(fact_10175_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(9)
thf(fact_10176_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N: num] :
                ( ( Xa2
                  = ( bit0 @ N ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N ) ) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa2
                    = ( bit1 @ N ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N ) ) ) ) )
           => ( ! [M3: num] :
                  ( ( X
                    = ( bit0 @ M3 ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M3 ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ! [N: num] :
                        ( ( Xa2
                          = ( bit0 @ N ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N: num] :
                          ( ( Xa2
                            = ( bit1 @ N ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit1 @ M3 ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M3 ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ! [N: num] :
                              ( ( Xa2
                                = ( bit0 @ N ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N ) ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N: num] :
                                ( ( Xa2
                                  = ( bit1 @ N ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_10177_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_10178_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).

% xor_num.simps(5)
thf(fact_10179_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_10180_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).

% xor_num.simps(9)
thf(fact_10181_xor__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ ( bit1 @ N2 ) ) ) ).

% xor_num.simps(2)
thf(fact_10182_xor__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ ( bit0 @ N2 ) ) ) ).

% xor_num.simps(3)
thf(fact_10183_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_10184_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_10185_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).

% xor_num.simps(8)
thf(fact_10186_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).

% xor_num.simps(6)
thf(fact_10187_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_10188_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G )
         => ( ( bfun_nat_real
              @ ^ [X2: nat] : ( F @ ( G @ X2 ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_10189_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).

% strict_mono_imp_increasing
thf(fact_10190_eventually__subseq,axiom,
    ! [R2: nat > nat,P: nat > $o] :
      ( ( order_5726023648592871131at_nat @ R2 )
     => ( ( eventually_nat @ P @ at_top_nat )
       => ( eventually_nat
          @ ^ [N3: nat] : ( P @ ( R2 @ N3 ) )
          @ at_top_nat ) ) ) ).

% eventually_subseq
thf(fact_10191_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F4: real > real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X3 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_10192_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10193_log__inj,axiom,
    ! [B: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_10194_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ N5 )
         => ( ord_less_eq_nat @ K @ N ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_10195_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_10196_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_10197_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_10198_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_10199_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ! [X3: nat] :
                ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X3 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_10200_open__bool__def,axiom,
    ( topolo9180104560040979295open_o
    = ( topolo4667128019001906403logy_o @ ( sup_sup_set_set_o @ ( image_o_set_o @ set_ord_lessThan_o @ top_top_set_o ) @ ( image_o_set_o @ set_or6416164934427428222Than_o @ top_top_set_o ) ) ) ) ).

% open_bool_def
thf(fact_10201_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_10202_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_10203_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_10204_open__nat__def,axiom,
    ( topolo4328251076210115529en_nat
    = ( topolo1613498594424996677gy_nat @ ( sup_sup_set_set_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) ) ) ) ).

% open_nat_def

% Helper facts (38)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

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 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X4: real] : ( P @ X4 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
    = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
      @ ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
          & ~ ( ( xa = mi )
              | ( xa = ma ) ) )
        @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ xa @ mi ) @ mi @ xa ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ) ).

%------------------------------------------------------------------------------