%------------------------------------------------------------------------------
% File     : ITP264^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DeleteCorrectness 01026_063622
% 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    : 0073_VEBT_DeleteCorrectness_01026_063622 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11207 (5646 unt; 954 typ;   0 def)
%            Number of atoms       : 28600 (12301 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 123613 (2769   ~; 523   |;1767   &;107416   @)
%                                         (   0 <=>;11138  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   92 (  91 usr)
%            Number of type conns  : 4134 (4134   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  866 ( 863 usr;  57 con; 0-8 aty)
%            Number of variables   : 26616 (2167   ^;23733   !; 716   ?;26616   :)
% 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 09:23:14.329
%------------------------------------------------------------------------------
% Could-be-implicit typings (91)
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_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_I_Eo_Mt__VEBT____Definitions__OVEBT_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__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__Product____Type__Oprod_I_Eo_Mt__Int__Oint_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__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__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__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__Code____Numeral__Ointeger,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__Complex__Ocomplex,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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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (863)
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_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_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,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_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,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

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,
    bit_se1409905431419307370or_int: int > int > int ).

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_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
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thf(sy_c_Complex_Ocomplex_ORe,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_GCD_Obezw__rel,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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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__Int__Oint,type,
    sup_sup_int: int > int > int ).

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_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_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

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_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

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

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_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_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__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_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

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_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__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__Int__Oint_J,type,
    nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).

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__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__Nat__Onat,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

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__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__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_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

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

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

thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: ( int > int > $o ) > list_int > $o ).

thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

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_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > 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__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > 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__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_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__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > 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__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__String__Ochar,type,
    size_size_char: char > 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_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_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,
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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,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
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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,
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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,
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thf(sy_c_Num_Onum_Osize__num,type,
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thf(sy_c_Num_Onum__of__nat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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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,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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thf(sy_c_Num_Opow,type,
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thf(sy_c_Num_Opred__numeral,type,
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thf(sy_c_Num_Osqr,type,
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thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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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,
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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,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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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__List__Olist_It__Nat__Onat_J_J,type,
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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,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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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,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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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__Int__Oint_J,type,
    ord_max_set_int: set_int > set_int > set_int ).

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_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: set_real > set_real > set_real ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > 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_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,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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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,
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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,
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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_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,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_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,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_Ofield__char__0__class_ORats_001t__Real__Oreal,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_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

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

thf(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: ( num > $o ) > set_num ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: ( rat > $o ) > set_rat ).

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

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__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,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

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

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

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

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

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__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

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_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_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_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_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_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_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_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_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__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__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_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $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_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $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__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $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_lx____,type,
    lx: 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_summin____,type,
    summin: nat ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10212)
thf(fact_0_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_1_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_2_bit__split__inv,axiom,
    ! [X2: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
      = X2 ) ).

% bit_split_inv
thf(fact_3__C4_Ohyps_C_I8_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "4.hyps"(8)
thf(fact_4__C12_C,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% "12"
thf(fact_5_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_6_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% high_def
thf(fact_7_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_8__C9_C,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% "9"
thf(fact_9_high__inv,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X2 ) @ N2 )
        = Y2 ) ) ).

% high_inv
thf(fact_10_low__inv,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X2 ) @ N2 )
        = X2 ) ) ).

% low_inv
thf(fact_11__C4_Ohyps_C_I7_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "4.hyps"(7)
thf(fact_12_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).

% bit_concat_def
thf(fact_13__C3_C,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "3"
thf(fact_14__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% \<open>summin * 2 ^ n + lx < 2 ^ deg\<close>
thf(fact_15_hprolist,axiom,
    ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) )
    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ).

% hprolist
thf(fact_16_False,axiom,
    ~ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ).

% False
thf(fact_17_nothprolist,axiom,
    ! [I: nat] :
      ( ( ( I
         != ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) )
        & ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ I )
        = ( nth_VEBT_VEBT @ treeList @ I ) ) ) ).

% nothprolist
thf(fact_18__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_092_060close_062,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ).

% \<open>length treeList = length (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)])\<close>
thf(fact_19_hlbound,axiom,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( ord_less_nat @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% hlbound
thf(fact_20_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_21__092_060open_062high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_060_Alength_AtreeList_092_060close_062,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).

% \<open>high (summin * 2 ^ n + lx) n < length treeList\<close>
thf(fact_22_notemp,axiom,
    ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ X_1 ) ).

% notemp
thf(fact_23_nnvalid,axiom,
    vEBT_invar_vebt @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ na ).

% nnvalid
thf(fact_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32__C6_C,axiom,
    ( ( ord_less_eq_nat @ mi @ ma )
    & ( ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ) ) ).

% "6"
thf(fact_33_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_34_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_V8194947554948674370ptions @ T @ Y2 ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_35__C8_C,axiom,
    na = m ).

% "8"
thf(fact_36__C2_C,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "2"
thf(fact_37_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_38_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_39_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_40_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_41_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_42__092_060open_062_092_060exists_062z_O_Aboth__member__options_A_ItreeList_A_B_Asummin_J_Az_092_060close_062,axiom,
    ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ summin ) @ X_1 ) ).

% \<open>\<exists>z. both_member_options (treeList ! summin) z\<close>
thf(fact_43__092_060open_062invar__vebt_A_ItreeList_A_B_Asummin_J_An_092_060close_062,axiom,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ summin ) @ na ).

% \<open>invar_vebt (treeList ! summin) n\<close>
thf(fact_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_num__double,axiom,
    ! [N2: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_73__092_060open_062summin_A_060_A2_A_094_Am_092_060close_062,axiom,
    ord_less_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).

% \<open>summin < 2 ^ m\<close>
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ 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: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
      = A2 ) ).

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

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_86_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ 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: real > $o,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

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

% Collect_cong
thf(fact_90_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_91__C1_C,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "1"
thf(fact_92__092_060open_062both__member__options_A_ItreeList_A_B_Ahigh_Ama_An_J_A_Ilow_Ama_An_J_092_060close_062,axiom,
    vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ma @ na ) ) @ ( vEBT_VEBT_low @ ma @ na ) ).

% \<open>both_member_options (treeList ! high ma n) (low ma n)\<close>
thf(fact_93__C7_C,axiom,
    ( ( mi != ma )
   => ! [I2: nat] :
        ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I2 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [Y3: nat] :
              ( ( ( ( vEBT_VEBT_high @ Y3 @ na )
                  = I2 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ Y3 @ na ) ) )
             => ( ( ord_less_nat @ mi @ Y3 )
                & ( ord_less_eq_nat @ Y3 @ ma ) ) ) ) ) ) ).

% "7"
thf(fact_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_yhelper,axiom,
    ! [Y2: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ Y2 @ na ) ) @ ( vEBT_VEBT_low @ Y2 @ na ) )
     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Y2 @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ord_less_nat @ mi @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ ma )
          & ( ord_less_nat @ ( vEBT_VEBT_low @ Y2 @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ) ) ).

% yhelper
thf(fact_105__C7b_C,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ( ( vEBT_VEBT_high @ ma @ na )
            = I2 )
         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
        & ! [Y3: nat] :
            ( ( ( ( vEBT_VEBT_high @ Y3 @ na )
                = I2 )
              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ Y3 @ na ) ) )
           => ( ( ord_less_nat @ mi @ Y3 )
              & ( ord_less_eq_nat @ Y3 @ ma ) ) ) ) ) ).

% "7b"
thf(fact_106_newlistlength,axiom,
    ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% newlistlength
thf(fact_107__C4_OIH_C_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X4 @ na )
        & ! [Xa: nat] : ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ X4 @ Xa ) @ na ) ) ) ).

% "4.IH"(1)
thf(fact_108__C4_C,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ X5 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).

% "4"
thf(fact_109__C5_C,axiom,
    ( ( mi = ma )
   => ! [X4: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ).

% "5"
thf(fact_110_le__num__One__iff,axiom,
    ! [X2: num] :
      ( ( ord_less_eq_num @ X2 @ one )
      = ( X2 = one ) ) ).

% le_num_One_iff
thf(fact_111_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L2: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_112_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L2: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_113_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_114_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_115_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_116_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_117_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_118_div__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_119_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_120_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_121_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_122_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_123_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_124_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_125_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_126_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_127_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_128_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_129_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_130_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_131_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_132_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_133_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_134_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_135_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_136_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_148_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_149_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_150_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_151_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_152_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_153_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_154_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_155_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_156_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_157_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% in_children_def
thf(fact_158__C111_C,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ I2 ) @ X5 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).

% "111"
thf(fact_159__C112_C,axiom,
    ( ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
          = ma )
       => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
          = ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) )
      & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
         != ma )
       => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
          = ma ) ) )
   => ! [X4: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ).

% "112"
thf(fact_160__092_060open_062mi_A_092_060noteq_062_Ama_A_092_060and_062_Ax_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    ( ( mi != ma )
    & ( ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ) ) ).

% \<open>mi \<noteq> ma \<and> x < 2 ^ deg\<close>
thf(fact_161_allvalidinlist,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
     => ( vEBT_invar_vebt @ X4 @ na ) ) ).

% allvalidinlist
thf(fact_162_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y2 ) @ X2 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_163_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_164_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_165_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_166_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_167_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_168_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_169_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X2: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_170_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_171_list__update__beyond,axiom,
    ! [Xs2: list_int,I: nat,X2: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( list_update_int @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_172_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_ding
thf(fact_173_sum__squares__bound,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_174_sum__squares__bound,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_175_power2__sum,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_176_power2__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( power_power_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_177_power2__sum,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_178_power2__sum,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_179_power2__sum,axiom,
    ! [X2: int,Y2: int] :
      ( ( power_power_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_180_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_181_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_182_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_183_inthall,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N2: nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
       => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_184_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_185_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_186_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_187_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_188_xmi,axiom,
    xa = mi ).

% xmi
thf(fact_189_list__update__overwrite,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ I @ Y2 )
      = ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ Y2 ) ) ).

% list_update_overwrite
thf(fact_190__C0_C,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ X4 @ na ) ) ).

% "0"
thf(fact_191__092_060open_062x_A_092_060noteq_062_Ami_A_092_060or_062_Ax_A_092_060noteq_062_Ama_092_060close_062,axiom,
    ( ( xa != mi )
    | ( xa != ma ) ) ).

% \<open>x \<noteq> mi \<or> x \<noteq> ma\<close>
thf(fact_192__C4_OIH_C_I2_J,axiom,
    ! [X2: nat] : ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ summary @ X2 ) @ m ) ).

% "4.IH"(2)
thf(fact_193_inrg,axiom,
    ( ( ord_less_eq_nat @ mi @ xa )
    & ( ord_less_eq_nat @ xa @ ma ) ) ).

% inrg
thf(fact_194__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
    vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).

% \<open>both_member_options summary (high ma n)\<close>
thf(fact_195_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_196_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X2: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_197_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X2: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_198_length__list__update,axiom,
    ! [Xs2: list_int,I: nat,X2: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I @ X2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_199_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_200_list__update__id,axiom,
    ! [Xs2: list_int,I: nat] :
      ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_201_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_202_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X2: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_203_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_int,X2: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ J )
        = ( nth_int @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_204_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B2 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_222_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B2 )
      = ( ! [X: complex] :
            ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_223_subset__code_I1_J,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B2 )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
           => ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_224_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B2 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_225_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B2 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_226_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_227_set__update__subsetI,axiom,
    ! [Xs2: list_real,A2: set_real,X2: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
     => ( ( member_real @ X2 @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_228_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A2: set_complex,X2: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
     => ( ( member_complex @ X2 @ A2 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_229_set__update__subsetI,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,I: nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
     => ( ( member8440522571783428010at_nat @ X2 @ A2 )
       => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_230_set__update__subsetI,axiom,
    ! [Xs2: list_int,A2: set_int,X2: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
     => ( ( member_int @ X2 @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_231_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBT @ X2 @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_232_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A2: set_nat,X2: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_nat @ X2 @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X2 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_233_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_234_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_235_four__x__squared,axiom,
    ! [X2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_236_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_237_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_238_nth__mem,axiom,
    ! [N2: nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
     => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N2 ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).

% nth_mem
thf(fact_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_in__set__conv__nth,axiom,
    ! [X2: real,Xs2: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_248_in__set__conv__nth,axiom,
    ! [X2: complex,Xs2: list_complex] :
      ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_249_in__set__conv__nth,axiom,
    ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_250_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_251_in__set__conv__nth,axiom,
    ! [X2: $o,Xs2: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_252_in__set__conv__nth,axiom,
    ! [X2: nat,Xs2: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_253_in__set__conv__nth,axiom,
    ! [X2: int,Xs2: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_254_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X2: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I4 ) ) )
     => ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_255_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X2: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I4 ) ) )
     => ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_256_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 ) ) )
     => ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_257_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_258_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X2: $o] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I4 ) ) )
     => ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_259_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X2: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I4 ) ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_260_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X2: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I4 ) ) )
     => ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_261_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_262_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_263_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_264_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_265_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_real,X2: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_266_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_complex,X2: complex] :
      ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X2 @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_267_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
     => ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_268_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_269_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_o,X2: $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_270_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_nat,X2: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_271_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_int,X2: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_272_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_273_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_274_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_275_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_276_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_277_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_278_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_279_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_280_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I5 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ I5 @ X6 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I5 @ X6 ) @ I @ X2 ) ) ) ).

% list_update_swap
thf(fact_281_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_282_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_283_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_284_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_285_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_286_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_287_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_288_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_289_power__commuting__commutes,axiom,
    ! [X2: complex,Y2: complex,N2: nat] :
      ( ( ( times_times_complex @ X2 @ Y2 )
        = ( times_times_complex @ Y2 @ X2 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X2 @ N2 ) @ Y2 )
        = ( times_times_complex @ Y2 @ ( power_power_complex @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_290_power__commuting__commutes,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( ( times_times_real @ X2 @ Y2 )
        = ( times_times_real @ Y2 @ X2 ) )
     => ( ( times_times_real @ ( power_power_real @ X2 @ N2 ) @ Y2 )
        = ( times_times_real @ Y2 @ ( power_power_real @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_291_power__commuting__commutes,axiom,
    ! [X2: nat,Y2: nat,N2: nat] :
      ( ( ( times_times_nat @ X2 @ Y2 )
        = ( times_times_nat @ Y2 @ X2 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X2 @ N2 ) @ Y2 )
        = ( times_times_nat @ Y2 @ ( power_power_nat @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_292_power__commuting__commutes,axiom,
    ! [X2: int,Y2: int,N2: nat] :
      ( ( ( times_times_int @ X2 @ Y2 )
        = ( times_times_int @ Y2 @ X2 ) )
     => ( ( times_times_int @ ( power_power_int @ X2 @ N2 ) @ Y2 )
        = ( times_times_int @ Y2 @ ( power_power_int @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_293_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_294_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_295_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_296_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_297_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_298_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
              = ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_309_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I4 )
              = ( nth_o @ Ys @ I4 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_310_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_311_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I4 )
              = ( nth_int @ Ys @ I4 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_312_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: vEBT_VEBT] : ( P @ I3 @ X5 ) ) )
      = ( ? [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_313_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: $o] : ( P @ I3 @ X5 ) ) )
      = ( ? [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_314_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: nat] : ( P @ I3 @ X5 ) ) )
      = ( ? [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_315_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X5: int] : ( P @ I3 @ X5 ) ) )
      = ( ? [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_316_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : Y4 = Z2 )
    = ( ^ [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_317_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_o,Z2: list_o] : Y4 = Z2 )
    = ( ^ [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_318_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_nat,Z2: list_nat] : Y4 = Z2 )
    = ( ^ [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_319_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_int,Z2: list_int] : Y4 = Z2 )
    = ( ^ [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_320_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_321_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X2: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X2 ) @ J )
        = ( ( ( I = J )
           => X2 )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_322_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_323_nth__list__update,axiom,
    ! [I: nat,Xs2: list_int,J: nat,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_324_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_325_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_326_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_327_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_328_power4__eq__xxxx,axiom,
    ! [X2: complex] :
      ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_329_power4__eq__xxxx,axiom,
    ! [X2: real] :
      ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_330_power4__eq__xxxx,axiom,
    ! [X2: nat] :
      ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_331_power4__eq__xxxx,axiom,
    ! [X2: int] :
      ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_341_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_342_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_343_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_344__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_061_A_Iif_Ax_A_061_Ami_Athen_Athe_A_Ivebt__mint_Asummary_J_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_Aelse_Ax_J_092_060close_062,axiom,
    ( ( ( xa = mi )
     => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
        = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
    & ( ( xa != mi )
     => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
        = xa ) ) ) ).

% \<open>summin * 2 ^ n + lx = (if x = mi then the (vebt_mint summary) * 2 ^ (deg div 2) + the (vebt_mint (treeList ! the (vebt_mint summary))) else x)\<close>
thf(fact_345_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_346_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_347_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_348_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y2 )
           => ( ( vEBT_vebt_member @ T @ Y2 )
              | ( X2 = Y2 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_349_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_350__092_060open_062vebt__member_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
    vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).

% \<open>vebt_member summary (high ma n)\<close>
thf(fact_351__092_060open_062vebt__member_A_ItreeList_A_B_Asummin_J_Alx_092_060close_062,axiom,
    vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ summin ) @ lx ).

% \<open>vebt_member (treeList ! summin) lx\<close>
thf(fact_352_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X2: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X2 )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_353_dsimp,axiom,
    ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ xa @ ma ) ) @ deg @ treeList @ summary ) @ xa )
    = ( vEBT_Node
      @ ( some_P7363390416028606310at_nat
        @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
          @ ( if_nat
            @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
              = ma )
            @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
            @ ma ) ) )
      @ deg
      @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
      @ summary ) ) ).

% dsimp
thf(fact_354_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_355_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_356_field__less__half__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_357_field__less__half__sum,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_358_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).

% min_Null_member
thf(fact_359_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
        = ( vEBT_vebt_member @ T @ X2 ) ) ) ).

% both_member_options_equiv_member
thf(fact_360_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
       => ( vEBT_vebt_member @ T @ X2 ) ) ) ).

% valid_member_both_member_options
thf(fact_361__C11_C,axiom,
    ord_less_eq_nat @ one_one_nat @ na ).

% "11"
thf(fact_362_semiring__norm_I87_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(87)
thf(fact_363_real__divide__square__eq,axiom,
    ! [R: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
      = ( divide_divide_real @ A @ R ) ) ).

% real_divide_square_eq
thf(fact_364_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_365_semiring__norm_I83_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit0 @ N2 ) ) ).

% semiring_norm(83)
thf(fact_366_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_367_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_368_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_369_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_370_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_371_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ one_one_real @ N2 )
      = one_one_real ) ).

% power_one
thf(fact_372_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_373_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ one_one_complex @ N2 )
      = one_one_complex ) ).

% power_one
thf(fact_374_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = 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 @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_375_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_376_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_377_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_378_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_379_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X2 )
        = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_380_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_381_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_382_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_383_semiring__norm_I12_J,axiom,
    ! [N2: num] :
      ( ( times_times_num @ one @ N2 )
      = N2 ) ).

% semiring_norm(12)
thf(fact_384_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_385_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_386_delt__out__of__range,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
        | ( ord_less_nat @ Ma @ X2 ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_387_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_388_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_389_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ 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_390_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ 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_391_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_392_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_393_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_394_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_395_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_396_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera6690914467698888265omplex @ N2 )
        = one_one_complex )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_397_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_real @ N2 )
        = one_one_real )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_398_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_399_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_400_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_401_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_402_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_403_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_404_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_405_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_406__C10_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).

% "10"
thf(fact_407_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( ( ord_less_nat @ X2 @ Ma )
            & ( ord_less_nat @ Mi @ X2 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_408_power__strict__increasing__iff,axiom,
    ! [B: real,X2: nat,Y2: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_409_power__strict__increasing__iff,axiom,
    ! [B: rat,X2: nat,Y2: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_410_power__strict__increasing__iff,axiom,
    ! [B: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_411_power__strict__increasing__iff,axiom,
    ! [B: int,X2: nat,Y2: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_412_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_413_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_414_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_415_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_416_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_417_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_418_power__increasing__iff,axiom,
    ! [B: real,X2: nat,Y2: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X2 ) @ ( power_power_real @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_419_power__increasing__iff,axiom,
    ! [B: rat,X2: nat,Y2: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X2 ) @ ( power_power_rat @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_420_power__increasing__iff,axiom,
    ! [B: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X2 ) @ ( power_power_nat @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_421_power__increasing__iff,axiom,
    ! [B: int,X2: nat,Y2: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X2 ) @ ( power_power_int @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_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_437_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_438_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_439_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_440_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_441_del__x__mi__lets__in__not__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_442_xnin,axiom,
    vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ).

% xnin
thf(fact_443__092_060open_062Some_Asummin_A_061_Avebt__mint_Asummary_092_060close_062,axiom,
    ( ( some_nat @ summin )
    = ( vEBT_vebt_mint @ summary ) ) ).

% \<open>Some summin = vebt_mint summary\<close>
thf(fact_444__092_060open_062Some_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_092_060close_062,axiom,
    ( ( some_nat @ lx )
    = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ summin ) ) ) ).

% \<open>Some lx = vebt_mint (treeList ! summin)\<close>
thf(fact_445_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X4: real] : ( member_real @ X4 @ S )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S )
           => ( ord_less_eq_real @ X3 @ Z3 ) )
       => ? [Y5: real] :
            ( ! [X4: real] :
                ( ( member_real @ X4 @ S )
               => ( ord_less_eq_real @ X4 @ Y5 ) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S )
                   => ( ord_less_eq_real @ X3 @ Z3 ) )
               => ( ord_less_eq_real @ Y5 @ Z3 ) ) ) ) ) ).

% complete_real
thf(fact_446_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_447_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_448_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_449_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_450_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_451_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_452_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_453_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_454_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_455_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_456_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_457_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_458_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_459_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_460_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_461_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_462_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_463_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X2 ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_464_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_465_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_466_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_467_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_468_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_469_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_470_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_471_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_472_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_473_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_474_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_475_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_476_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_477_left__right__inverse__power,axiom,
    ! [X2: rat,Y2: rat,N2: nat] :
      ( ( ( times_times_rat @ X2 @ Y2 )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X2 @ N2 ) @ ( power_power_rat @ Y2 @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_478_left__right__inverse__power,axiom,
    ! [X2: complex,Y2: complex,N2: nat] :
      ( ( ( times_times_complex @ X2 @ Y2 )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ N2 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_479_left__right__inverse__power,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( ( times_times_real @ X2 @ Y2 )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_480_left__right__inverse__power,axiom,
    ! [X2: nat,Y2: nat,N2: nat] :
      ( ( ( times_times_nat @ X2 @ Y2 )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_481_left__right__inverse__power,axiom,
    ! [X2: int,Y2: int,N2: nat] :
      ( ( ( times_times_int @ X2 @ Y2 )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_482_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_483_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_484_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_485_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_486_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_487_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_488_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_489_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_490_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_491_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_492_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_493_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_494_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_495_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_496_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_497_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_498_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_499_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_500_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_501_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_502_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_503_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_504_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_505_power__increasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_increasing
thf(fact_506_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_507_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_508_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_509_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_510_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_511_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_512_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_513_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_514_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_515_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_516_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_517_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 )
       => ? [N4: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_518_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 )
       => ? [N4: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_519_field__sum__of__halves,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X2 ) ).

% field_sum_of_halves
thf(fact_520_field__sum__of__halves,axiom,
    ! [X2: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X2 ) ).

% field_sum_of_halves
thf(fact_521_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: 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 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [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 )
                         => ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( 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 @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_522_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: 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 @ TreeList2 @ 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_523_insert__simp__norm,axiom,
    ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_524_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X2: 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 @ TreeList2 ) )
     => ( ( ord_less_nat @ X2 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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_525_del__single__cont,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( X2 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_even__odd__cases,axiom,
    ! [X2: nat] :
      ( ! [N4: nat] :
          ( X2
         != ( plus_plus_nat @ N4 @ N4 ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).

% even_odd_cases
thf(fact_534_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_535_maxbmo,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X2 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).

% maxbmo
thf(fact_536_power__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( power_power_nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_537__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062summin_O_ASome_Asummin_A_061_Avebt__mint_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Summin: nat] :
        ( ( some_nat @ Summin )
       != ( vEBT_vebt_mint @ summary ) ) ).

% \<open>\<And>thesis. (\<And>summin. Some summin = vebt_mint summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_538_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_539_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_540_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_541_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_542_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_543_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Mini: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X2 )
         => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).

% mint_corr_help
thf(fact_544_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X2 )
         => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_545__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lx_O_ASome_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Lx: nat] :
        ( ( some_nat @ Lx )
       != ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ summin ) ) ) ).

% \<open>\<And>thesis. (\<And>lx. Some lx = vebt_mint (treeList ! summin) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_546_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_547_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_548_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_549_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_550_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_551_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_552_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_553_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_554_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_555_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_556_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_557_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_558_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(5)
thf(fact_559_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X2 ) )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(5)
thf(fact_560_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(5)
thf(fact_561_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(5)
thf(fact_562_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(5)
thf(fact_563_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(5)
thf(fact_564_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(6)
thf(fact_565_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(6)
thf(fact_566_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ one_one_real )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(6)
thf(fact_567_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(6)
thf(fact_568_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(6)
thf(fact_569_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ one_one_int )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(6)
thf(fact_570_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_571_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_572_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_573_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_574_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_575_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_576_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_577_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_578_Suc__inject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% Suc_inject
thf(fact_579_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N2: extended_enat] :
      ( ! [N4: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N4 )
             => ( P @ M2 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% enat_less_induct
thf(fact_580_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_581_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_582_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_583_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_584_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_585_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_586_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_587_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_588_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_589_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_590_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_591_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_592_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_593_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_594_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M3: nat] :
            ( ( M
              = ( suc @ M3 ) )
            & ( ord_less_nat @ N2 @ M3 ) ) ) ) ).

% Suc_less_eq2
thf(fact_595_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_596_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_597_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_598_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
       => ( ! [I4: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I4 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I4 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I4 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_599_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I4: nat] :
            ( ( J
              = ( suc @ I4 ) )
           => ( P @ I4 ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ J )
             => ( ( P @ ( suc @ I4 ) )
               => ( P @ I4 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_600_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_601_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_602_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_603_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_604_Suc__le__D,axiom,
    ! [N2: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_605_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_606_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_607_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_608_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N4 )
             => ( P @ M2 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_609_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ M @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_610_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y5: nat,Z4: nat] :
              ( ( R2 @ X3 @ Y5 )
             => ( ( R2 @ Y5 @ Z4 )
               => ( R2 @ X3 @ Z4 ) ) )
         => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
           => ( R2 @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_611_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_612_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_613_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_614_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_615_real__arch__pow,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ? [N4: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N4 ) ) ) ).

% real_arch_pow
thf(fact_616_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_617_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_618_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_619_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_620_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_621_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_622_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_623_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_624_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_625_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_626_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_627_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_628_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_629_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_630_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_631_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_632_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_633_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_634_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_635_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_636_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_637_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_638_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ N4 )
                 => ( P @ ( suc @ N4 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_639_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ ( suc @ N4 ) )
                 => ( P @ N4 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_640_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_641_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_642_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_643_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_644_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_645_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q3: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_646_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_647_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_648_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_649_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_650_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_651_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_652_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_653_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_654_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_655_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_656_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_657_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_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_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_670_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_671_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_672_less__not__refl3,axiom,
    ! [S3: nat,T: nat] :
      ( ( ord_less_nat @ S3 @ T )
     => ( S3 != T ) ) ).

% less_not_refl3
thf(fact_673_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_674_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N4 )
             => ( P @ M2 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_675_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N4 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_676_linorder__neqE__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_nat @ X2 @ Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_677_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_678_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_679_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_680_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_681_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_682_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y3: nat] :
                ( ( P @ Y3 )
               => ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_683_size__neq__size__imp__neq,axiom,
    ! [X2: list_VEBT_VEBT,Y2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X2 )
       != ( size_s6755466524823107622T_VEBT @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_684_size__neq__size__imp__neq,axiom,
    ! [X2: list_o,Y2: list_o] :
      ( ( ( size_size_list_o @ X2 )
       != ( size_size_list_o @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_685_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ X2 )
       != ( size_size_list_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_686_size__neq__size__imp__neq,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( ( size_size_list_int @ X2 )
       != ( size_size_list_int @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_687_size__neq__size__imp__neq,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( size_size_num @ X2 )
       != ( size_size_num @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_688_div__nat__eqI,axiom,
    ! [N2: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_689_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_690_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( 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 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_691_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_692_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_693_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_694_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_695_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_696_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N: nat] :
          ( ( ord_less_eq_nat @ M6 @ N )
          & ( M6 != N ) ) ) ) ).

% nat_less_le
thf(fact_697_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_698_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N: nat] :
          ( ( ord_less_nat @ M6 @ N )
          | ( M6 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_699_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_700_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_701_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I4: nat,J2: nat] :
          ( ( ord_less_nat @ I4 @ J2 )
         => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_702_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_703_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L2 )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_less_mono
thf(fact_704_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_705_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_706_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_707_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_708_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_709_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_710_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_711_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_712_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_713_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_714_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_715_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N4: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_716_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_le_mono
thf(fact_717_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_718_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_719_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_720_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_721_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_722_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_723_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).

% mult_le_mono
thf(fact_724_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_725_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_726_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_727_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_728_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_729_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_730_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( 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 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_731_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_nat @ M5 @ N4 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_732_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: 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 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [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 )
                         => ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( 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 @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_733_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).

% maxt_corr
thf(fact_734_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X2 ) ) ) ) ).

% maxt_sound
thf(fact_735_mint__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X2 ) ) ) ) ).

% mint_sound
thf(fact_736_mint__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).

% mint_corr
thf(fact_737_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_738_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_739_succ__min,axiom,
    ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_740_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_741_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_742_helpyd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Y2 ) )
       => ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpyd
thf(fact_743_helpypredd,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Y2 ) )
       => ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpypredd
thf(fact_744_succ__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T @ Y2 )
        & ( ord_less_nat @ X2 @ Y2 )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ X2 @ Z5 ) )
           => ( ord_less_eq_nat @ Y2 @ Z5 ) ) ) ) ).

% succ_member
thf(fact_745_pred__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T @ Y2 )
        & ( ord_less_nat @ Y2 @ X2 )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ Z5 @ X2 ) )
           => ( ord_less_eq_nat @ Z5 @ Y2 ) ) ) ) ).

% pred_member
thf(fact_746_pred__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).

% pred_corr
thf(fact_747_succ__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% succ_corr
thf(fact_748_pred__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% pred_correct
thf(fact_749_succ__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% succ_correct
thf(fact_750_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_751_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M5: nat] :
              ( ( ( some_nat @ M5 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_752_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_753_pred__list__to__short,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_754_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_755_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_756_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_757_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_758_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_759_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_760_not__None__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( X2 != none_P5556105721700978146at_nat )
      = ( ? [Y: product_prod_nat_nat] :
            ( X2
            = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_761_not__None__eq,axiom,
    ! [X2: option_nat] :
      ( ( X2 != none_nat )
      = ( ? [Y: nat] :
            ( X2
            = ( some_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_762_not__None__eq,axiom,
    ! [X2: option_num] :
      ( ( X2 != none_num )
      = ( ? [Y: num] :
            ( X2
            = ( some_num @ Y ) ) ) ) ).

% not_None_eq
thf(fact_763_not__Some__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( ! [Y: product_prod_nat_nat] :
            ( X2
           != ( some_P7363390416028606310at_nat @ Y ) ) )
      = ( X2 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_764_not__Some__eq,axiom,
    ! [X2: option_nat] :
      ( ( ! [Y: nat] :
            ( X2
           != ( some_nat @ Y ) ) )
      = ( X2 = none_nat ) ) ).

% not_Some_eq
thf(fact_765_not__Some__eq,axiom,
    ! [X2: option_num] :
      ( ( ! [Y: num] :
            ( X2
           != ( some_num @ Y ) ) )
      = ( X2 = none_num ) ) ).

% not_Some_eq
thf(fact_766_max__less__iff__conj,axiom,
    ! [X2: extended_enat,Y2: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X2 @ Y2 ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X2 @ Z )
        & ( ord_le72135733267957522d_enat @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_767_max__less__iff__conj,axiom,
    ! [X2: code_integer,Y2: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X2 @ Y2 ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X2 @ Z )
        & ( ord_le6747313008572928689nteger @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_768_max__less__iff__conj,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_real @ X2 @ Z )
        & ( ord_less_real @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_769_max__less__iff__conj,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_rat @ X2 @ Z )
        & ( ord_less_rat @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_770_max__less__iff__conj,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_num @ X2 @ Z )
        & ( ord_less_num @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_771_max__less__iff__conj,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_nat @ X2 @ Z )
        & ( ord_less_nat @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_772_max__less__iff__conj,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_int @ X2 @ Z )
        & ( ord_less_int @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_773_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_774_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_775_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_776_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_777_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_778_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_779_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_780_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_781_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_782_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_783_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_784_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_785_max_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
      = ( ord_max_nat @ A @ B ) ) ).

% max.right_idem
thf(fact_786_max_Oright__idem,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ ( ord_ma741700101516333627d_enat @ A @ B ) @ B )
      = ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.right_idem
thf(fact_787_max_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
      = ( ord_max_int @ A @ B ) ) ).

% max.right_idem
thf(fact_788_max_Oright__idem,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_max_Code_integer @ A @ B ) @ B )
      = ( ord_max_Code_integer @ A @ B ) ) ).

% max.right_idem
thf(fact_789_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
      = ( ord_max_nat @ A @ B ) ) ).

% max.left_idem
thf(fact_790_max_Oleft__idem,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) )
      = ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.left_idem
thf(fact_791_max_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
      = ( ord_max_int @ A @ B ) ) ).

% max.left_idem
thf(fact_792_max_Oleft__idem,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_max_Code_integer @ A @ ( ord_max_Code_integer @ A @ B ) )
      = ( ord_max_Code_integer @ A @ B ) ) ).

% max.left_idem
thf(fact_793_max_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ A )
      = A ) ).

% max.idem
thf(fact_794_max_Oidem,axiom,
    ! [A: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ A @ A )
      = A ) ).

% max.idem
thf(fact_795_max_Oidem,axiom,
    ! [A: int] :
      ( ( ord_max_int @ A @ A )
      = A ) ).

% max.idem
thf(fact_796_max_Oidem,axiom,
    ! [A: code_integer] :
      ( ( ord_max_Code_integer @ A @ A )
      = A ) ).

% max.idem
thf(fact_797_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_798_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_eq_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_799_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_800_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_801_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_802_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_803_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_804_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_805_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_806_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_807_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_808_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_809_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_810_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_811_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_812_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_813_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_814_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
      = ( ( ord_le3102999989581377725nteger @ B @ A )
        & ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_815_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_816_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_817_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_818_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_819_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_820_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_821_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_822_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_823_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_824_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_825_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_848_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_849_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_850_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_851_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_852_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_853_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_854_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_855_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_856_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N2 @ L2 ) ) ) ).

% diff_le_mono
thf(fact_857_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_858_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_859_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_860_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_861_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: num > num > num,Uv: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_862_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_863_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_864_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_865_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_866_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_867_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_868_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_869_mult__diff__mult,axiom,
    ! [X2: rat,Y2: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X2 @ Y2 ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X2 @ ( minus_minus_rat @ Y2 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_870_mult__diff__mult,axiom,
    ! [X2: complex,Y2: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X2 @ Y2 ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X2 @ ( minus_minus_complex @ Y2 @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_871_mult__diff__mult,axiom,
    ! [X2: real,Y2: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X2 @ Y2 ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X2 @ ( minus_minus_real @ Y2 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_872_mult__diff__mult,axiom,
    ! [X2: int,Y2: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X2 @ Y2 ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X2 @ ( minus_minus_int @ Y2 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X2 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_873_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y2: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y2 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y2 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B3 ) )
                   => ( Y2
                     != ( some_P7363390416028606310at_nat @ ( X2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_893_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y2: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ( ( Xa2 = none_num )
         => ( Y2 != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y2 != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2
                  = ( some_num @ A3 ) )
               => ! [B3: num] :
                    ( ( Xb
                      = ( some_num @ B3 ) )
                   => ( Y2
                     != ( some_num @ ( X2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_894_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y2: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ( ( Xa2 = none_nat )
         => ( Y2 != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y2 != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2
                  = ( some_nat @ A3 ) )
               => ! [B3: nat] :
                    ( ( Xb
                      = ( some_nat @ B3 ) )
                   => ( Y2
                     != ( some_nat @ ( X2 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_903_max_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ C ) )
      = ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_904_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.left_commute
thf(fact_905_max_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_max_Code_integer @ B @ ( ord_max_Code_integer @ A @ C ) )
      = ( ord_max_Code_integer @ A @ ( ord_max_Code_integer @ B @ C ) ) ) ).

% max.left_commute
thf(fact_906_max_Ocommute,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_max_nat @ B4 @ A4 ) ) ) ).

% max.commute
thf(fact_907_max_Ocommute,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A4: extended_enat,B4: extended_enat] : ( ord_ma741700101516333627d_enat @ B4 @ A4 ) ) ) ).

% max.commute
thf(fact_908_max_Ocommute,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( ord_max_int @ B4 @ A4 ) ) ) ).

% max.commute
thf(fact_909_max_Ocommute,axiom,
    ( ord_max_Code_integer
    = ( ^ [A4: code_integer,B4: code_integer] : ( ord_max_Code_integer @ B4 @ A4 ) ) ) ).

% max.commute
thf(fact_910_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.assoc
thf(fact_911_max_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ ( ord_ma741700101516333627d_enat @ A @ B ) @ C )
      = ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ B @ C ) ) ) ).

% max.assoc
thf(fact_912_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.assoc
thf(fact_913_max_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_max_Code_integer @ ( ord_max_Code_integer @ A @ B ) @ C )
      = ( ord_max_Code_integer @ A @ ( ord_max_Code_integer @ B @ C ) ) ) ).

% max.assoc
thf(fact_914_power2__commute,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_915_power2__commute,axiom,
    ! [X2: real,Y2: real] :
      ( ( power_power_real @ ( minus_minus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_916_power2__commute,axiom,
    ! [X2: int,Y2: int] :
      ( ( power_power_int @ ( minus_minus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_917_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_918_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_919_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_920_power2__diff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_921_power2__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( power_power_real @ ( minus_minus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_922_power2__diff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_923_power2__diff,axiom,
    ! [X2: int,Y2: int] :
      ( ( power_power_int @ ( minus_minus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_924_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_925_max_Omono,axiom,
    ! [C: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A )
     => ( ( ord_le3102999989581377725nteger @ D @ B )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C @ D ) @ ( ord_max_Code_integer @ A @ B ) ) ) ) ).

% max.mono
thf(fact_926_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_927_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_928_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_929_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_930_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_931_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( A
        = ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.orderE
thf(fact_932_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_933_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_934_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_935_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_936_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_937_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( ord_max_Code_integer @ A @ B ) )
     => ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% max.orderI
thf(fact_938_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_939_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_940_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_941_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_942_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_943_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le3102999989581377725nteger @ B @ A )
         => ~ ( ord_le3102999989581377725nteger @ C @ A ) ) ) ).

% max.boundedE
thf(fact_944_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_945_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_946_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_947_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_948_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_949_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ A )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_950_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_951_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_952_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_953_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_954_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B4: extended_enat,A4: extended_enat] :
          ( A4
          = ( ord_ma741700101516333627d_enat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_955_max_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B4: code_integer,A4: code_integer] :
          ( A4
          = ( ord_max_Code_integer @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_956_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( A4
          = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_957_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( A4
          = ( ord_max_num @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_958_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_959_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( A4
          = ( ord_max_int @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_960_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_961_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded1
thf(fact_962_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_963_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_964_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_965_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_966_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_967_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : ( ord_le3102999989581377725nteger @ B @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded2
thf(fact_968_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_969_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_970_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_971_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_972_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X2: extended_enat,Y2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y2 ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X2 )
        | ( ord_le2932123472753598470d_enat @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_973_le__max__iff__disj,axiom,
    ! [Z: code_integer,X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Z @ ( ord_max_Code_integer @ X2 @ Y2 ) )
      = ( ( ord_le3102999989581377725nteger @ Z @ X2 )
        | ( ord_le3102999989581377725nteger @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_974_le__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y2 ) )
      = ( ( ord_less_eq_rat @ Z @ X2 )
        | ( ord_less_eq_rat @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_975_le__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y2: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y2 ) )
      = ( ( ord_less_eq_num @ Z @ X2 )
        | ( ord_less_eq_num @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_976_le__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ( ord_less_eq_nat @ Z @ X2 )
        | ( ord_less_eq_nat @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_977_le__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ( ord_less_eq_int @ Z @ X2 )
        | ( ord_less_eq_int @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_978_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B4: extended_enat,A4: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_979_max_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B4: code_integer,A4: code_integer] :
          ( ( ord_max_Code_integer @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_980_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_981_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_982_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_983_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_984_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A4: extended_enat,B4: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_985_max_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A4: code_integer,B4: code_integer] :
          ( ( ord_max_Code_integer @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_986_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_987_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_988_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_989_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_990_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_991_max_OcoboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_992_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_993_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_994_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_995_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_996_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_997_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ B )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_998_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_999_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_1000_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_1001_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_1002_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X2: extended_enat,Y2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y2 ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X2 )
        | ( ord_le72135733267957522d_enat @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1003_less__max__iff__disj,axiom,
    ! [Z: code_integer,X2: code_integer,Y2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X2 @ Y2 ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X2 )
        | ( ord_le6747313008572928689nteger @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1004_less__max__iff__disj,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y2 ) )
      = ( ( ord_less_real @ Z @ X2 )
        | ( ord_less_real @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1005_less__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y2 ) )
      = ( ( ord_less_rat @ Z @ X2 )
        | ( ord_less_rat @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1006_less__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y2: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y2 ) )
      = ( ( ord_less_num @ Z @ X2 )
        | ( ord_less_num @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1007_less__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ Z @ X2 )
        | ( ord_less_nat @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1008_less__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ( ord_less_int @ Z @ X2 )
        | ( ord_less_int @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_1009_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_1010_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1011_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_1012_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_1013_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_1014_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_1015_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_1016_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B4: extended_enat,A4: extended_enat] :
          ( ( A4
            = ( ord_ma741700101516333627d_enat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1017_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B4: code_integer,A4: code_integer] :
          ( ( A4
            = ( ord_max_Code_integer @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1018_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( A4
            = ( ord_max_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1019_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( A4
            = ( ord_max_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1020_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( A4
            = ( ord_max_num @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1021_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( ord_max_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1022_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( A4
            = ( ord_max_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1023_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_1024_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_1032_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_1033_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_1034_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_1035_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_1036_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_1037_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1038_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1039_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y2
                  = ( some_num @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1040_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X2
                = ( some_nat @ A3 ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1041_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X2
                = ( some_nat @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1042_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: nat,B3: num] :
              ( ( X2
                = ( some_nat @ A3 ) )
             => ( ( Y2
                  = ( some_num @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1043_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X2
                = ( some_num @ A3 ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1044_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: num,B3: nat] :
              ( ( X2
                = ( some_num @ A3 ) )
             => ( ( Y2
                  = ( some_nat @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1045_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A3: num,B3: num] :
              ( ( X2
                = ( some_num @ A3 ) )
             => ( ( Y2
                  = ( some_num @ B3 ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_1046_split__option__all,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ! [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_1047_split__option__all,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ! [X7: option_nat] : ( P2 @ X7 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_1048_split__option__all,axiom,
    ( ( ^ [P2: option_num > $o] :
        ! [X7: option_num] : ( P2 @ X7 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_1049_split__option__ex,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ? [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_1050_split__option__ex,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ? [X7: option_nat] : ( P2 @ X7 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_1051_split__option__ex,axiom,
    ( ( ^ [P2: option_num > $o] :
        ? [X7: option_num] : ( P2 @ X7 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_1052_option_Oexhaust,axiom,
    ! [Y2: option4927543243414619207at_nat] :
      ( ( Y2 != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y2
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1053_option_Oexhaust,axiom,
    ! [Y2: option_nat] :
      ( ( Y2 != none_nat )
     => ~ ! [X23: nat] :
            ( Y2
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1054_option_Oexhaust,axiom,
    ! [Y2: option_num] :
      ( ( Y2 != none_num )
     => ~ ! [X23: num] :
            ( Y2
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_1055_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_1056_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_1057_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_1058_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_1059_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_1060_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_1061_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_1062_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_1063_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_1064_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
        = ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( ( the_Pr8591224930841456533at_nat @ Option )
              = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_1065_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
        = ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( ( the_nat @ Option )
              = ( the_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_1066_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
        = ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( ( the_num @ Option )
              = ( the_num @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_1067_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1068_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1069_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_1079_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_1080_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_1081_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_1082_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_1083_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_1084_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1085_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1086_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1087_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1088_add__diff__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1089_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1090_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1091_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1092_diff__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1093_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1094_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_add__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_1106_mul__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( times_times_nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_1132_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_1133_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_1134_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1135_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_1136_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_1137_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_1138_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_1139_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1140_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y2: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y2 )
     => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y2 @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y2 ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_1152_linorder__neqE__linordered__idom,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ Y2 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1153_linorder__neqE__linordered__idom,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_rat @ X2 @ Y2 )
       => ( ord_less_rat @ Y2 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1154_linorder__neqE__linordered__idom,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_int @ X2 @ Y2 )
       => ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1155_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_1156_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_1157_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_1158_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_1159_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_1160_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_1161_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_1162_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_1163_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1164_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_1165_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_1166_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_1167_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1168_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_1169_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_1170_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_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_1184_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_1185_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_1186_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1198_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1199_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1200_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1201_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_1202_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_1203_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_1204_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_1205_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1206_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1207_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1208_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1209_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_1210_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_1211_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_1212_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_1213_one__reorient,axiom,
    ! [X2: complex] :
      ( ( one_one_complex = X2 )
      = ( X2 = one_one_complex ) ) ).

% one_reorient
thf(fact_1214_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
      = ( X2 = one_one_real ) ) ).

% one_reorient
thf(fact_1215_one__reorient,axiom,
    ! [X2: rat] :
      ( ( one_one_rat = X2 )
      = ( X2 = one_one_rat ) ) ).

% one_reorient
thf(fact_1216_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_1217_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
      = ( X2 = one_one_int ) ) ).

% one_reorient
thf(fact_1218_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_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C2: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_1227_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_1228_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_1229_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_1230_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_1231_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1241_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1242_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1243_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1244_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1245_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1246_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1247_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1248_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1249_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1250_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1251_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1273_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1274_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1275_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1276_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1277_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1278_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1279_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1280_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1281_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1282_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1283_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_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_1294_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_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_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_1306_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_1307_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_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_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_1322_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_1323_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_1324_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_1325_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_1326_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_1327_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_1328_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_1329_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_1330_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_1331_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_1332_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_1333_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_1334_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_1335_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_1336_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1337_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1338_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1339_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1340_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1341_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_1342_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_1343_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_1344_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_1345_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_1346_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1347_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_1348_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_1349_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_1350_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1351_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_1352_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_1353_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_1354_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1355_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_1356_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_1357_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1358_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_1359_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_1360_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_1361_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_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_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_1375_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_1376_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_1377_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_1378_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_1379_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_1380_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_1381_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_1382_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_1383_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_1384_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_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_max__add__distrib__right,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y2 @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1399_max__add__distrib__right,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y2 @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1400_max__add__distrib__right,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y2 @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1401_max__add__distrib__right,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y2 @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1402_max__add__distrib__right,axiom,
    ! [X2: code_integer,Y2: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X2 @ ( ord_max_Code_integer @ Y2 @ Z ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X2 @ Y2 ) @ ( plus_p5714425477246183910nteger @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1403_max__add__distrib__left,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y2 ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1404_max__add__distrib__left,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y2 ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1405_max__add__distrib__left,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y2 ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1406_max__add__distrib__left,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y2 ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1407_max__add__distrib__left,axiom,
    ! [X2: code_integer,Y2: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X2 @ Y2 ) @ Z )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X2 @ Z ) @ ( plus_p5714425477246183910nteger @ Y2 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1408_max__diff__distrib__left,axiom,
    ! [X2: code_integer,Y2: code_integer,Z: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X2 @ Y2 ) @ Z )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ Z ) @ ( minus_8373710615458151222nteger @ Y2 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_1409_max__diff__distrib__left,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y2 ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y2 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_1410_max__diff__distrib__left,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y2 ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y2 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_1411_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_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1420_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1421_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1422_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1423_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1424_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1425_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1426_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1427_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_1428_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_1429_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_1430_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_1431_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_1432_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_1433_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_1434_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_1435_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_1436_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_1437_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_1438_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_1439_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_1440_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_1441_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_1442_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_1443_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_1444_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_1445_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_1446_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_1447_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_1448_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_1449_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_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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_1467_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_1468_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_1469_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_1470_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_1471_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_1472_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_1473_square__diff__square__factored,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( minus_minus_rat @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_1474_square__diff__square__factored,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X2 @ X2 ) @ ( times_times_complex @ Y2 @ Y2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ X2 @ Y2 ) @ ( minus_minus_complex @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_1475_square__diff__square__factored,axiom,
    ! [X2: real,Y2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
      = ( times_times_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_real @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_1476_square__diff__square__factored,axiom,
    ! [X2: int,Y2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
      = ( times_times_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( minus_minus_int @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_1477_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_1478_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_1479_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_1480_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_1481_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_1482_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_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_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_1490_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_1491_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_1492_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_1493_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_1494_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_1495_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_1496_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_1497_square__diff__one__factored,axiom,
    ! [X2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_1498_square__diff__one__factored,axiom,
    ! [X2: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X2 @ X2 ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X2 @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_1499_square__diff__one__factored,axiom,
    ! [X2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_1500_square__diff__one__factored,axiom,
    ! [X2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_1501_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_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_divmod__step__eq,axiom,
    ! [L2: num,R: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_1518_divmod__step__eq,axiom,
    ! [L2: num,R: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_1519_divmod__step__eq,axiom,
    ! [L2: num,R: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_1520_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_1521_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_1522_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_1523_del__x__not__mi,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_1524_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_1525_del__x__not__mia,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X2 = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_1526_del__in__range,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_1527_del__x__mi,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_1528_del__x__mi__lets__in,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_1529_del__x__mi__lets__in__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_1530_del__x__mia,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_1531_pred__less__length__list,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_1532_pred__lesseq__max,axiom,
    ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_1533_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_1534_mult__commute__abs,axiom,
    ! [C: complex] :
      ( ( ^ [X: complex] : ( times_times_complex @ X @ C ) )
      = ( times_times_complex @ C ) ) ).

% mult_commute_abs
thf(fact_1535_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_1536_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_1537_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_1538_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1539_lambda__one,axiom,
    ( ( ^ [X: complex] : X )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_1540_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_1541_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1542_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1543_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( 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] :
            ( X2
           != ( 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,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( X2
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1544_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X2
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X2
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
              ( X2
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1545_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X2
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X2
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A3: num,B3: num] :
              ( X2
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1546_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X2
           != ( 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,Y5: product_prod_nat_nat] :
              ( X2
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1547_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X2
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X2
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X3: nat,Y5: nat] :
              ( X2
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1548_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X2
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X2
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X3: num,Y5: num] :
              ( X2
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1549_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_1550_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_1551_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_1552_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_1553_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_1554_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_1555_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_1556_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_1557_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_1558_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_1559_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y5: real] : ( ord_less_real @ Y5 @ X4 ) ).

% linordered_field_no_lb
thf(fact_1560_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y5: rat] : ( ord_less_rat @ Y5 @ X4 ) ).

% linordered_field_no_lb
thf(fact_1561_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1562_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1563_times__divide__times__eq,axiom,
    ! [X2: complex,Y2: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X2 @ Y2 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ Y2 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1564_times__divide__times__eq,axiom,
    ! [X2: real,Y2: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1565_times__divide__times__eq,axiom,
    ! [X2: rat,Y2: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1566_divide__divide__times__eq,axiom,
    ! [X2: complex,Y2: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X2 @ Y2 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ W ) @ ( times_times_complex @ Y2 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1567_divide__divide__times__eq,axiom,
    ! [X2: real,Y2: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1568_divide__divide__times__eq,axiom,
    ! [X2: rat,Y2: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X2 @ W ) @ ( times_times_rat @ Y2 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1569_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_1570_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_1571_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_1572_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_1573_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_1574_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_1575_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_1576_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_1577_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_1578_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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 ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_1579_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_1580_vebt__succ_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_1581_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs )
          & ( ord_less_nat @ Y @ X )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ Z5 @ X )
               => ( ord_less_eq_nat @ Z5 @ Y ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_1582_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs )
          & ( ord_less_nat @ X @ Y )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ X @ Z5 )
               => ( ord_less_eq_nat @ Y @ Z5 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_1583_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1584_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1585_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_1586_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_1587_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_1588_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_1589_vebt__delete_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X2 @ Mi )
          | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X2 @ Mi )
            | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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 ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_1590_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
      = ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ ( ord_less_nat @ X2 @ Mi )
            & ( ~ ( ord_less_nat @ X2 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X2 )
                & ( ~ ( ord_less_nat @ Ma @ X2 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_1591_succ__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T @ Y )
                & ( ord_less_nat @ X2 @ Y ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_1592_pred__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T @ Y )
                & ( ord_less_nat @ Y @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_1593_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_1594_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S3 ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_1595_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_1596_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_1597_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_1598_vebt__pred_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y2 != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y2
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y2 = none_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va2 ) ) )
                 => ~ ( ( B3
                       => ( Y2
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A3
                           => ( Y2
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y2 = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y2 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
                 => ( Y2 != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                   => ( Y2 != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y2
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y2
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_1599_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1600_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1601_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_1602_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1603_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A4: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_1604_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B3: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_1605_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1606_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X )
          | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_1607_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X2 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
          | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1608_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_1609_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1610_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1611_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_1612_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_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_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_1626_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1627_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_1628_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1629_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1630_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1631_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1632_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_1633_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1634_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1635_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1636_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_1637_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1638_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1639_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1640_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1641_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y2 ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1642_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y2 )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_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_1653_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_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1664_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1665_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1666_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_1667_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1668_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1669_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1670_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1671_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1672_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1673_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1674_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1675_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1676_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1677_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1678_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1679_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1680_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1681_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1682_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1683_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1684_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1685_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1686_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1687_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1688_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_1689_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_1690_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_1691_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_1692_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_1693_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_1694_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_1695_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_1696_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_1697_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1698_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_1699_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_1700_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1701_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1702_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1703_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_1704_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_1705_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_1706_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1707_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_1708_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1709_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1710_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_1711_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_1712_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_1713_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_1714_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_1715_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_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_1727_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_1728_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_1729_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_1730_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_1731_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_1732_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_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_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_1742_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_1743_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_1744_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_1745_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_1746_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_1747_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_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_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_1757_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_1758_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_1759_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_1760_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_1761_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_1762_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_1763_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_1764_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_1765_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_1766_sum__squares__eq__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) )
        = zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1767_sum__squares__eq__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1768_sum__squares__eq__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1769_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_1770_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_1771_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_1772_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_1773_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_1774_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_1775_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_1776_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_1777_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_1778_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_1779_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_1780_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_1781_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_1782_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_1783_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_1784_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_1785_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_1786_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_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_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_1796_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_1797_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_1798_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_1799_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_1800_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_1801_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_1802_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_1803_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_1804_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_1805_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_1806_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_1807_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_1808_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_1809_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_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1818_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1819_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1820_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1821_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1822_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1823_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1824_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1825_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1826_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_1827_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_1828_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_1829_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_1830_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_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1838_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1839_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1840_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_1841_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_1842_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_1843_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_1844_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_1845_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_1846_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1847_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1848_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1849_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1850_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1851_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1852_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1853_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1854_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1855_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1856_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1857_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1858_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1859_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1860_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1861_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_1862_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(4)
thf(fact_1863_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(4)
thf(fact_1864_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(4)
thf(fact_1865_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(4)
thf(fact_1866_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(4)
thf(fact_1867_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(4)
thf(fact_1868_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(3)
thf(fact_1869_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X2 ) )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(3)
thf(fact_1870_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(3)
thf(fact_1871_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(3)
thf(fact_1872_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(3)
thf(fact_1873_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(3)
thf(fact_1874_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_1875_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_1876_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_1877_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_1878_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_1879_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_1880_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_1881_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_1882_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_1883_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_1884_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_1885_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_1886_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_1887_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_1888_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_1889_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1890_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_1891_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_1892_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_1893_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_1894_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1895_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M: nat] :
      ( ( ( power_power_nat @ X2 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1896_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1897_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_1898_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_1899_nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1900_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_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_1912_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_1913_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_1914_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_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_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_1932_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_1933_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_1934_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_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_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_1960_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_1961_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_1962_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_1963_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_1964_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_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_power2__eq__iff__nonneg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1976_power2__eq__iff__nonneg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1977_power2__eq__iff__nonneg,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1978_power2__eq__iff__nonneg,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_sum__power2__eq__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1990_sum__power2__eq__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1991_sum__power2__eq__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1992_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_1993_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_1994_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_1995_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1996_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_1997_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1998_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1999_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_2000_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_2001_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_2002_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X2
           != ( 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] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2003_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_2004_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_2005_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_2006_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_2007_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_2008_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_2009_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_2010_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_2011_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_2012_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_2013_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_2014_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_2015_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y2
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y2
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_2016_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_2017_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A: $o,B: $o] :
      ( ( ( X2 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X2 != zero_zero_nat )
       => ( ( ( X2 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X2 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X2 )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_2018_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_2019_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_2020_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2021_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2022_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2023_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2024_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_2025_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2026_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2027_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_2028_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_2029_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_2030_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_2031_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_2032_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_2033_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_2034_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2035_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2036_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2037_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2038_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_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_2058_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_2059_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_2060_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_2061_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_2062_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_2063_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_2064_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2065_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2066_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2067_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2068_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_2069_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2070_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2071_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2072_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2073_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_2074_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_2075_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_2076_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_2077_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_2078_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2079_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2080_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2081_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2082_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2083_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_2084_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_2085_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_2086_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_2087_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_2088_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_2089_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_2090_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_2091_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_2092_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_2093_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_2094_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y2
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_2095_vebt__buildup_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( X2
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va2: nat] :
              ( X2
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_2096_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_2097_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
       => ( ! [X3: nat,Y5: nat] :
              ( ( P @ X3 @ Y5 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_2098_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_2099_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_2100_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_2101_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_2102_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_2103_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N4 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_2104_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_2105_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_2106_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_2107_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_2108_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_2109_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_2110_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2111_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_2112_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_2113_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_2114_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_2115_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_2116_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_2117_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2118_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_2119_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_2120_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N2 ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2130_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_2131_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_2132_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2133_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2134_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_2135_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_2136_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_2137_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_2138_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_2139_vebt__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
            ( X2
           != ( 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] :
              ( X2
             != ( 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] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_2140_vebt__delete_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B3: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,N4: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N4 ) ) ) )
         => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X3 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_2141_VEBT__internal_OminNull_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( X2
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_2142_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_2143_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_2144_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_2145_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2146_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_2147_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2148_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2149_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2150_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2151_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_2152_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_2153_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_2154_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_2155_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_2156_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_2157_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_2158_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_2159_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_2160_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_2161_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_2162_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_2163_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_2164_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_2165_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_2166_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_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_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_2180_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_2181_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_2182_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_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_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_2201_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_2202_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_2203_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_2204_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_2205_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2206_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2207_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2208_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_2209_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_2210_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_2211_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_2212_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_2213_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_2214_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_2215_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_2216_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_2217_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_2218_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_2219_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_2220_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_2221_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_2222_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_2223_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_2224_add__nonpos__eq__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X2 @ Y2 )
            = zero_zero_real )
          = ( ( X2 = zero_zero_real )
            & ( Y2 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2225_add__nonpos__eq__0__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X2 @ Y2 )
            = zero_zero_rat )
          = ( ( X2 = zero_zero_rat )
            & ( Y2 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2226_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X2 @ Y2 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2227_add__nonpos__eq__0__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X2 @ Y2 )
            = zero_zero_int )
          = ( ( X2 = zero_zero_int )
            & ( Y2 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2228_add__nonneg__eq__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( plus_plus_real @ X2 @ Y2 )
            = zero_zero_real )
          = ( ( X2 = zero_zero_real )
            & ( Y2 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2229_add__nonneg__eq__0__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ( plus_plus_rat @ X2 @ Y2 )
            = zero_zero_rat )
          = ( ( X2 = zero_zero_rat )
            & ( Y2 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2230_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ( ( plus_plus_nat @ X2 @ Y2 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2231_add__nonneg__eq__0__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ( plus_plus_int @ X2 @ Y2 )
            = zero_zero_int )
          = ( ( X2 = zero_zero_int )
            & ( Y2 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2232_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_2233_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_2234_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_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_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_2242_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_2243_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_2244_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_2245_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_2246_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_2247_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_2248_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_2249_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_2250_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_2251_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_2252_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_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_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_2260_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_2261_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_2262_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_2263_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_2264_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2265_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2266_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2267_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2268_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_2269_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_2270_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_2271_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_2272_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_2273_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_2274_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_2275_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_2276_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_2277_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_2278_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_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_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_2292_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_2293_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_2294_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_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_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_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_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_2314_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_2315_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_2316_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_2317_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_2318_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_2319_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_2320_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_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2329_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2330_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2331_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_2332_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_2333_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_2334_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( X2
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2335_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_2336_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_2337_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_2338_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_2339_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_add__less__zeroD,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
        | ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2349_add__less__zeroD,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X2 @ zero_zero_rat )
        | ( ord_less_rat @ Y2 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2350_add__less__zeroD,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ zero_zero_int )
     => ( ( ord_less_int @ X2 @ zero_zero_int )
        | ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2351_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2352_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2353_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2354_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2355_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2356_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2357_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2358_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2359_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2360_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2361_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2362_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2363_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2364_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2365_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2366_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_2367_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_2368_divide__nonpos__nonpos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2369_divide__nonpos__nonpos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2370_divide__nonpos__nonneg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2371_divide__nonpos__nonneg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2372_divide__nonneg__nonpos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2373_divide__nonneg__nonpos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2374_divide__nonneg__nonneg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2375_divide__nonneg__nonneg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2376_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_2377_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_2378_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_2379_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_2380_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_2381_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_2382_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2383_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2384_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_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_2399_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_2400_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_2401_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_2402_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_2403_divide__pos__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_pos_pos
thf(fact_2404_divide__pos__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_pos_pos
thf(fact_2405_divide__pos__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2406_divide__pos__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2407_divide__neg__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2408_divide__neg__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2409_divide__neg__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_neg_neg
thf(fact_2410_divide__neg__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_neg_neg
thf(fact_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_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_2420_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_2421_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_2422_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_2423_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_2424_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_2425_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_2426_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_2427_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_2428_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_2429_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_2430_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_2431_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_2432_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_2433_frac__eq__eq,axiom,
    ! [Y2: complex,Z: complex,X2: complex,W: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X2 @ Y2 )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X2 @ Z )
            = ( times_times_complex @ W @ Y2 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2434_frac__eq__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X2 @ Y2 )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X2 @ Z )
            = ( times_times_real @ W @ Y2 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2435_frac__eq__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X2 @ Y2 )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X2 @ Z )
            = ( times_times_rat @ W @ Y2 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2436_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_2437_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_2438_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_2439_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2440_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2441_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2442_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2443_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2444_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_2445_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M6: nat] :
            ( N2
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2446_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_2447_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_2448_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_2449_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_2450_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_2451_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_2452_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2453_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2454_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2455_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2456_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2457_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2458_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2459_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_2460_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_2461_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_2462_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_2463_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_2464_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2465_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_2466_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_2467_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_2468_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_2469_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).

% vebt_member.simps(3)
thf(fact_2470_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X2
               != ( 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] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2471_vebt__insert_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X3 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_2472_vebt__succ_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B3: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve ) )
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_2473_vebt__pred_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw2: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,Va2: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
             => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_2474_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ X2 )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) ) ).

% vebt_insert.simps(2)
thf(fact_2475_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_2476_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y2 )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y2 )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y2 )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y2 )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_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_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_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_2506_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_2507_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_2508_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_2509_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_2510_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_2511_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_2512_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_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_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_2520_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_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_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_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_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_2536_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_2537_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_2538_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_2539_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_2540_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_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_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_2549_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_2550_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_2551_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_2552_field__le__epsilon,axiom,
    ! [X2: real,Y2: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ E2 ) ) )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% field_le_epsilon
thf(fact_2553_field__le__epsilon,axiom,
    ! [X2: rat,Y2: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y2 @ E2 ) ) )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% field_le_epsilon
thf(fact_2554_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_2555_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_2556_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_2557_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_2558_divide__nonpos__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2559_divide__nonpos__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2560_divide__nonpos__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2561_divide__nonpos__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2562_divide__nonneg__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2563_divide__nonneg__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2564_divide__nonneg__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2565_divide__nonneg__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2566_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_2567_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_2568_frac__less2,axiom,
    ! [X2: real,Y2: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2569_frac__less2,axiom,
    ! [X2: rat,Y2: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ Y2 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2570_frac__less,axiom,
    ! [X2: real,Y2: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2571_frac__less,axiom,
    ! [X2: rat,Y2: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ X2 @ Y2 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2572_frac__le,axiom,
    ! [Y2: real,X2: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2573_frac__le,axiom,
    ! [Y2: rat,X2: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ X2 @ Y2 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2574_sum__squares__ge__zero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2575_sum__squares__ge__zero,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2576_sum__squares__ge__zero,axiom,
    ! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2577_sum__squares__le__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2578_sum__squares__le__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2579_sum__squares__le__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2580_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_2581_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_2582_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_2583_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_2584_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_2585_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_2586_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_2587_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_2588_mult__right__le__one__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y2 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2589_mult__right__le__one__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y2 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2590_mult__right__le__one__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ord_less_eq_int @ Y2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y2 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2591_mult__left__le__one__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y2 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2592_mult__left__le__one__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y2 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2593_mult__left__le__one__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ord_less_eq_int @ Y2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y2 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2594_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_2595_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_2596_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_2597_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_2598_not__sum__squares__lt__zero,axiom,
    ! [X2: real,Y2: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2599_not__sum__squares__lt__zero,axiom,
    ! [X2: rat,Y2: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2600_not__sum__squares__lt__zero,axiom,
    ! [X2: int,Y2: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2601_sum__squares__gt__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) )
      = ( ( X2 != zero_zero_real )
        | ( Y2 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2602_sum__squares__gt__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) )
      = ( ( X2 != zero_zero_rat )
        | ( Y2 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2603_sum__squares__gt__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) )
      = ( ( X2 != zero_zero_int )
        | ( Y2 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2604_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_2605_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_2606_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2607_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2608_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2609_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2610_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_2611_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_2612_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_2613_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_2614_mult__imp__less__div__pos,axiom,
    ! [Y2: real,Z: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y2 ) @ X2 )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2615_mult__imp__less__div__pos,axiom,
    ! [Y2: rat,Z: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y2 ) @ X2 )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2616_mult__imp__div__pos__less,axiom,
    ! [Y2: real,X2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y2 ) )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2617_mult__imp__div__pos__less,axiom,
    ! [Y2: rat,X2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2618_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_2619_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_2620_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_2621_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_2622_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_2623_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_2624_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_2625_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_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_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_2637_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_2638_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_2639_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_2640_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_2641_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_2642_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_2643_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_2644_divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2645_divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y2 )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2646_divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2647_add__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y2 @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2648_add__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y2 @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2649_add__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y2 @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2650_add__num__frac,axiom,
    ! [Y2: complex,Z: complex,X2: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y2 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_num_frac
thf(fact_2651_add__num__frac,axiom,
    ! [Y2: real,Z: real,X2: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_num_frac
thf(fact_2652_add__num__frac,axiom,
    ! [Y2: rat,Z: rat,X2: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_num_frac
thf(fact_2653_add__frac__num,axiom,
    ! [Y2: complex,X2: complex,Z: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y2 ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_frac_num
thf(fact_2654_add__frac__num,axiom,
    ! [Y2: real,X2: real,Z: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_frac_num
thf(fact_2655_add__frac__num,axiom,
    ! [Y2: rat,X2: rat,Z: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_frac_num
thf(fact_2656_add__frac__eq,axiom,
    ! [Y2: complex,Z: complex,X2: complex,W: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y2 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y2 ) ) @ ( times_times_complex @ Y2 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2657_add__frac__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2658_add__frac__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2659_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_2660_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_2661_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_2662_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_2663_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_2664_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_2665_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_2666_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_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_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_2675_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_2676_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_2677_divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y2 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2678_divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y2 )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2679_divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y2 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2680_diff__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y2 @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2681_diff__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y2 @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2682_diff__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y2 @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2683_diff__frac__eq,axiom,
    ! [Y2: complex,Z: complex,X2: complex,W: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y2 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y2 ) ) @ ( times_times_complex @ Y2 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2684_diff__frac__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2685_diff__frac__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2686_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_2687_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_2688_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_2689_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).

% vebt_member.simps(4)
thf(fact_2690_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) ).

% vebt_delete.simps(5)
thf(fact_2691_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_2692_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_2693_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_2694_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_2695_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_2696_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_2697_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_2698_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_2699_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2700_length__pos__if__in__set,axiom,
    ! [X2: real,Xs2: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2701_length__pos__if__in__set,axiom,
    ! [X2: complex,Xs2: list_complex] :
      ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2702_length__pos__if__in__set,axiom,
    ! [X2: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2703_length__pos__if__in__set,axiom,
    ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2704_length__pos__if__in__set,axiom,
    ! [X2: $o,Xs2: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2705_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs2: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2706_length__pos__if__in__set,axiom,
    ! [X2: int,Xs2: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2707_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_2708_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_2709_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_2710_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_2711_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_2712_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_2713_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_2714_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2715_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_2716_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_2717_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X2 )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) ).

% vebt_insert.simps(3)
thf(fact_2718_vebt__mint_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X2
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_2719_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) ).

% vebt_delete.simps(6)
thf(fact_2720_vebt__mint_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y2
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y2
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y2 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2721_vebt__maxt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y2
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y2
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y2 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y2 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y2
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2722_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_2723_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_2724_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_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_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_2732_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_2733_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_2734_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_2735_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_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_field__le__mult__one__interval,axiom,
    ! [X2: real,Y2: real] :
      ( ! [Z4: real] :
          ( ( ord_less_real @ zero_zero_real @ Z4 )
         => ( ( ord_less_real @ Z4 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X2 ) @ Y2 ) ) )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% field_le_mult_one_interval
thf(fact_2747_field__le__mult__one__interval,axiom,
    ! [X2: rat,Y2: rat] :
      ( ! [Z4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z4 )
         => ( ( ord_less_rat @ Z4 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X2 ) @ Y2 ) ) )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% field_le_mult_one_interval
thf(fact_2748_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_2749_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_2750_mult__imp__le__div__pos,axiom,
    ! [Y2: real,Z: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y2 ) @ X2 )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2751_mult__imp__le__div__pos,axiom,
    ! [Y2: rat,Z: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y2 ) @ X2 )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2752_mult__imp__div__pos__le,axiom,
    ! [Y2: real,X2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y2 ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2753_mult__imp__div__pos__le,axiom,
    ! [Y2: rat,X2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2754_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_2755_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_2756_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_2757_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_2758_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_2759_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_2760_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_2761_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_2762_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_2763_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_2764_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_2765_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_2766_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_2767_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_2768_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_2769_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_2770_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_2771_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_2772_convex__bound__le,axiom,
    ! [X2: real,A: real,Y2: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X2 @ A )
     => ( ( ord_less_eq_real @ Y2 @ 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 @ X2 ) @ ( times_times_real @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2773_convex__bound__le,axiom,
    ! [X2: rat,A: rat,Y2: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X2 @ A )
     => ( ( ord_less_eq_rat @ Y2 @ 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 @ X2 ) @ ( times_times_rat @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2774_convex__bound__le,axiom,
    ! [X2: int,A: int,Y2: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X2 @ A )
     => ( ( ord_less_eq_int @ Y2 @ 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 @ X2 ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2775_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_2776_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_2777_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_2778_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_2779_frac__le__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2780_frac__le__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2781_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_2782_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_2783_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_2784_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_2785_frac__less__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2786_frac__less__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2787_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_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_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_2795_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2796_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2797_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2798_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2799_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2800_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2801_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2802_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2803_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2804_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2805_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2806_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2807_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2808_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_2809_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_2810_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_2811_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_2812_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_2813_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_2814_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_2815_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_2816_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_2817_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2818_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_2819_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_2820_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_2821_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_2822_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_2823_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M6 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_2824_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_2825_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_2826_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_2827_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2828_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_2829_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_2830_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_2831_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_2832_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
        = Y2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y2
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y2 )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( Y2
                  = ( ~ ( ( ( 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_2833_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X2
                  = ( 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_2834_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X2
                    = ( 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_2835_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_2836_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve2 )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_2837_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_2838_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y2 )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y2 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y2
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y2
                    = ( ~ ( ( 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] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( Y2
                      = ( ~ ( ( ( 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_2839_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X2
                      = ( 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] :
                        ( X2
                        = ( 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_2840_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_2841_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_2842_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_2843_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_2844_convex__bound__lt,axiom,
    ! [X2: real,A: real,Y2: real,U: real,V: real] :
      ( ( ord_less_real @ X2 @ A )
     => ( ( ord_less_real @ Y2 @ 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 @ X2 ) @ ( times_times_real @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2845_convex__bound__lt,axiom,
    ! [X2: rat,A: rat,Y2: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X2 @ A )
     => ( ( ord_less_rat @ Y2 @ 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 @ X2 ) @ ( times_times_rat @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2846_convex__bound__lt,axiom,
    ! [X2: int,A: int,Y2: int,U: int,V: int] :
      ( ( ord_less_int @ X2 @ A )
     => ( ( ord_less_int @ Y2 @ 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 @ X2 ) @ ( times_times_int @ V @ Y2 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2847_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_2848_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_2849_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_2850_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_2851_scaling__mono,axiom,
    ! [U: real,V: real,R: real,S3: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R )
       => ( ( ord_less_eq_real @ R @ S3 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2852_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S3: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
       => ( ( ord_less_eq_rat @ R @ S3 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2853_power2__le__imp__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_2854_power2__le__imp__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_2855_power2__le__imp__le,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_2856_power2__le__imp__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_2857_power2__eq__imp__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2858_power2__eq__imp__eq,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2859_power2__eq__imp__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2860_power2__eq__imp__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_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_2868_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_2869_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_2870_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_2871_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_2872_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_2873_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_2874_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_2875_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_2876_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_2877_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_2878_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P4: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2879_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P4: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2880_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P4: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2881_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P4: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2882_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P4: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2883_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_2884_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_2885_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_2886_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_2887_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_2888_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_2889_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_2890_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_2891_power2__less__imp__less,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_2892_power2__less__imp__less,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_rat @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_2893_power2__less__imp__less,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_2894_power2__less__imp__less,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_int @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_2895_sum__power2__le__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2896_sum__power2__le__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2897_sum__power2__le__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2898_sum__power2__ge__zero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2899_sum__power2__ge__zero,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2900_sum__power2__ge__zero,axiom,
    ! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2901_sum__power2__gt__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_real )
        | ( Y2 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2902_sum__power2__gt__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_rat )
        | ( Y2 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2903_sum__power2__gt__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_int )
        | ( Y2 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2904_not__sum__power2__lt__zero,axiom,
    ! [X2: real,Y2: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2905_not__sum__power2__lt__zero,axiom,
    ! [X2: rat,Y2: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_2906_not__sum__power2__lt__zero,axiom,
    ! [X2: int,Y2: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2907_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_2908_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_2909_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_2910_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_2911_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_2912_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_2913_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_2914_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
        = Y2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y2
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y2 )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y2 )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y2 )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y2
                      = ( ~ ( ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_2915_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_2916_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_2917_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_2918_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_2919_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_2920_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_2921_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_2922_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( 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 @ X2 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_2923_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( 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 @ X2 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_2924_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X2
                  = ( 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] :
                    ( X2
                    = ( 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_2925_arith__geo__mean,axiom,
    ! [U: real,X2: real,Y2: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X2 @ Y2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2926_arith__geo__mean,axiom,
    ! [U: rat,X2: rat,Y2: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X2 @ Y2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2927_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A4: $o,B4: $o] :
                ( A1
                = ( vEBT_Leaf @ A4 @ B4 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_2928_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X4 @ N4 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                     => ( ( M5 = N4 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N4 @ M5 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X4 @ N4 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                       => ( ( M5
                            = ( suc @ N4 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N4 @ M5 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X4 @ N4 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                         => ( ( M5 = N4 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N4 @ M5 ) )
                             => ( ! [I2: nat] :
                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I2: nat] :
                                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                    = I2 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                & ! [X4: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
                                                        = I2 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X4 )
                                                      & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M5: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X4 @ N4 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M5 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                           => ( ( M5
                                = ( suc @ N4 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N4 @ M5 ) )
                               => ( ! [I2: nat] :
                                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I2: nat] :
                                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                      = I2 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                  & ! [X4: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
                                                          = I2 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X4 )
                                                        & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_2929_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
        = Y2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y2
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y2
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y2
                        = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
             => ( Y2
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
               => ( Y2
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y2
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y2
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_2930_vebt__delete_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
        = Y2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y2
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y2
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [N4: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N4 ) ) )
                 => ( Y2
                   != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( Y2
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                   => ( Y2
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                     => ( Y2
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y2
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y2
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y2
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va2 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va2 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_2931_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_2932_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_2933_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).

% vebt_member.simps(2)
thf(fact_2934_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_2935_vebt__succ_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
        = Y2 )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( X2
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y2
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y2 = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa2
                  = ( suc @ N4 ) )
             => ( Y2 != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y2 != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y2 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                 => ( Y2 != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y2
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y2
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_2936_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_2937_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_2938_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_2939_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_2940_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_2941_max__bot,axiom,
    ! [X2: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X2 )
      = X2 ) ).

% max_bot
thf(fact_2942_max__bot,axiom,
    ! [X2: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X2 )
      = X2 ) ).

% max_bot
thf(fact_2943_max__bot,axiom,
    ! [X2: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X2 )
      = X2 ) ).

% max_bot
thf(fact_2944_max__bot,axiom,
    ! [X2: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X2 )
      = X2 ) ).

% max_bot
thf(fact_2945_max__bot,axiom,
    ! [X2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X2 )
      = X2 ) ).

% max_bot
thf(fact_2946_max__bot2,axiom,
    ! [X2: set_nat] :
      ( ( ord_max_set_nat @ X2 @ bot_bot_set_nat )
      = X2 ) ).

% max_bot2
thf(fact_2947_max__bot2,axiom,
    ! [X2: set_int] :
      ( ( ord_max_set_int @ X2 @ bot_bot_set_int )
      = X2 ) ).

% max_bot2
thf(fact_2948_max__bot2,axiom,
    ! [X2: set_real] :
      ( ( ord_max_set_real @ X2 @ bot_bot_set_real )
      = X2 ) ).

% max_bot2
thf(fact_2949_max__bot2,axiom,
    ! [X2: nat] :
      ( ( ord_max_nat @ X2 @ bot_bot_nat )
      = X2 ) ).

% max_bot2
thf(fact_2950_max__bot2,axiom,
    ! [X2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ X2 @ bot_bo4199563552545308370d_enat )
      = X2 ) ).

% max_bot2
thf(fact_2951_vebt__succ_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B3
                     => ( Y2
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y2 = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa2
                      = ( suc @ N4 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2 = 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] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y2 = 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,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                     => ( ( Y2 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y2
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y2
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_2952_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_2953_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_2954_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_2955_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_2956_subset__empty,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_2957_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_2958_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_2959_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_2960_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_2961_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_min_max
thf(fact_2962_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_leaf
thf(fact_2963_order__refl,axiom,
    ! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_2964_order__refl,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).

% order_refl
thf(fact_2965_order__refl,axiom,
    ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).

% order_refl
thf(fact_2966_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_2967_order__refl,axiom,
    ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).

% order_refl
thf(fact_2968_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_2969_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_2970_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_2971_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_2972_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_2973_subset__antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% subset_antisym
thf(fact_2974_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% psubsetI
thf(fact_2975_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( member_real @ X3 @ B2 ) )
     => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% subsetI
thf(fact_2976_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( member_int @ X3 @ B2 ) )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% subsetI
thf(fact_2977_subsetI,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( member_complex @ X3 @ B2 ) )
     => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).

% subsetI
thf(fact_2978_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A2 )
         => ( member8440522571783428010at_nat @ X3 @ B2 ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_2979_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( member_nat @ X3 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_2980_div__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_2981_div__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_2982_i0__less,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_2983_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_2984_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = N2 ) ).

% idiff_0_right
thf(fact_2985_not__real__square__gt__zero,axiom,
    ! [X2: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
      = ( X2 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_2986_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_2987_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_2988_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_2989_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_2990_int__div__less__self,axiom,
    ! [X2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).

% int_div_less_self
thf(fact_2991_div__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_2992_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_2993_not__iless0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_2994_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_2995_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).

% i0_lb
thf(fact_2996_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2997_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_2998_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_2999_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3000_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_3001_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_3002_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_3003_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_3004_le__cases3,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X2 @ Y2 )
       => ~ ( ord_less_eq_rat @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y2 @ X2 )
         => ~ ( ord_less_eq_rat @ X2 @ Z ) )
       => ( ( ( ord_less_eq_rat @ X2 @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y2 )
             => ~ ( ord_less_eq_rat @ Y2 @ X2 ) )
           => ( ( ( ord_less_eq_rat @ Y2 @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X2 )
                 => ~ ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3005_le__cases3,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ( ord_less_eq_num @ X2 @ Y2 )
       => ~ ( ord_less_eq_num @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_num @ Y2 @ X2 )
         => ~ ( ord_less_eq_num @ X2 @ Z ) )
       => ( ( ( ord_less_eq_num @ X2 @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_num @ Z @ Y2 )
             => ~ ( ord_less_eq_num @ Y2 @ X2 ) )
           => ( ( ( ord_less_eq_num @ Y2 @ Z )
               => ~ ( ord_less_eq_num @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_num @ Z @ X2 )
                 => ~ ( ord_less_eq_num @ X2 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3006_le__cases3,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y2 )
       => ~ ( ord_less_eq_nat @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y2 @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y2 )
             => ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y2 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3007_le__cases3,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ( ord_less_eq_int @ X2 @ Y2 )
       => ~ ( ord_less_eq_int @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_int @ Y2 @ X2 )
         => ~ ( ord_less_eq_int @ X2 @ Z ) )
       => ( ( ( ord_less_eq_int @ X2 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_int @ Z @ Y2 )
             => ~ ( ord_less_eq_int @ Y2 @ X2 ) )
           => ( ( ( ord_less_eq_int @ Y2 @ Z )
               => ~ ( ord_less_eq_int @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X2 )
                 => ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3008_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3009_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3010_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: num,Z2: num] : Y4 = Z2 )
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3011_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3012_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3013_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3014_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3015_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3016_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3017_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3018_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3019_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3020_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3021_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3022_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3023_order__antisym,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_set_nat @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_3024_order__antisym,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ( ord_less_eq_rat @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_3025_order__antisym,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
     => ( ( ord_less_eq_num @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_3026_order__antisym,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_3027_order__antisym,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_3028_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_3029_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_3030_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_3031_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_3032_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_3033_order__trans,axiom,
    ! [X2: set_nat,Y2: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_set_nat @ Y2 @ Z )
       => ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_3034_order__trans,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ( ord_less_eq_rat @ Y2 @ Z )
       => ( ord_less_eq_rat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_3035_order__trans,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
     => ( ( ord_less_eq_num @ Y2 @ Z )
       => ( ord_less_eq_num @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_3036_order__trans,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z )
       => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_3037_order__trans,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z )
       => ( ord_less_eq_int @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_3038_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: rat,B3: rat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3039_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: num,B3: num] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3040_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3041_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3042_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3043_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3044_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: num,Z2: num] : Y4 = Z2 )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3045_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3046_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3047_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_3048_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_3049_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_3050_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_3051_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_3052_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3053_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3054_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3055_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3056_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3057_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3058_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3059_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3060_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3061_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3062_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3063_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : Y4 = Z2 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3064_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: num,Z2: num] : Y4 = Z2 )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3065_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3066_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3067_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3068_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3069_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3070_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_eq_int @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3071_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3072_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3073_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3074_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_eq_int @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3075_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3076_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3077_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3078_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3079_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3080_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3081_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3082_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3083_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3084_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3085_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3086_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3087_order__eq__refl,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_3088_order__eq__refl,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_3089_order__eq__refl,axiom,
    ! [X2: num,Y2: num] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_num @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_3090_order__eq__refl,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_3091_order__eq__refl,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_3092_linorder__linear,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
      | ( ord_less_eq_rat @ Y2 @ X2 ) ) ).

% linorder_linear
thf(fact_3093_linorder__linear,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
      | ( ord_less_eq_num @ Y2 @ X2 ) ) ).

% linorder_linear
thf(fact_3094_linorder__linear,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
      | ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% linorder_linear
thf(fact_3095_linorder__linear,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
      | ( ord_less_eq_int @ Y2 @ X2 ) ) ).

% linorder_linear
thf(fact_3096_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3097_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3098_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3099_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3100_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3101_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3102_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3103_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3104_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3105_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3106_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3107_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3108_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3109_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3110_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3111_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3112_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3113_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3114_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3115_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3116_linorder__le__cases,axiom,
    ! [X2: rat,Y2: rat] :
      ( ~ ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ord_less_eq_rat @ Y2 @ X2 ) ) ).

% linorder_le_cases
thf(fact_3117_linorder__le__cases,axiom,
    ! [X2: num,Y2: num] :
      ( ~ ( ord_less_eq_num @ X2 @ Y2 )
     => ( ord_less_eq_num @ Y2 @ X2 ) ) ).

% linorder_le_cases
thf(fact_3118_linorder__le__cases,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% linorder_le_cases
thf(fact_3119_linorder__le__cases,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_eq_int @ X2 @ Y2 )
     => ( ord_less_eq_int @ Y2 @ X2 ) ) ).

% linorder_le_cases
thf(fact_3120_order__antisym__conv,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ X2 )
     => ( ( ord_less_eq_set_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_3121_order__antisym__conv,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y2 @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_3122_order__antisym__conv,axiom,
    ! [Y2: num,X2: num] :
      ( ( ord_less_eq_num @ Y2 @ X2 )
     => ( ( ord_less_eq_num @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_3123_order__antisym__conv,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_3124_order__antisym__conv,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ Y2 @ X2 )
     => ( ( ord_less_eq_int @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_3125_lt__ex,axiom,
    ! [X2: real] :
    ? [Y5: real] : ( ord_less_real @ Y5 @ X2 ) ).

% lt_ex
thf(fact_3126_lt__ex,axiom,
    ! [X2: rat] :
    ? [Y5: rat] : ( ord_less_rat @ Y5 @ X2 ) ).

% lt_ex
thf(fact_3127_lt__ex,axiom,
    ! [X2: int] :
    ? [Y5: int] : ( ord_less_int @ Y5 @ X2 ) ).

% lt_ex
thf(fact_3128_gt__ex,axiom,
    ! [X2: real] :
    ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).

% gt_ex
thf(fact_3129_gt__ex,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_3130_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_3131_gt__ex,axiom,
    ! [X2: int] :
    ? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).

% gt_ex
thf(fact_3132_dense,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ? [Z4: real] :
          ( ( ord_less_real @ X2 @ Z4 )
          & ( ord_less_real @ Z4 @ Y2 ) ) ) ).

% dense
thf(fact_3133_dense,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ? [Z4: rat] :
          ( ( ord_less_rat @ X2 @ Z4 )
          & ( ord_less_rat @ Z4 @ Y2 ) ) ) ).

% dense
thf(fact_3134_less__imp__neq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_3135_less__imp__neq,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_3136_less__imp__neq,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_3137_less__imp__neq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_3138_less__imp__neq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% less_imp_neq
thf(fact_3139_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_3140_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_3141_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_3142_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_3143_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_3144_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3145_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3146_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3147_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3148_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3149_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3150_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3151_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3152_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3153_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3154_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y3: nat] :
              ( ( ord_less_nat @ Y3 @ X3 )
             => ( P @ Y3 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_3155_antisym__conv3,axiom,
    ! [Y2: real,X2: real] :
      ( ~ ( ord_less_real @ Y2 @ X2 )
     => ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_3156_antisym__conv3,axiom,
    ! [Y2: rat,X2: rat] :
      ( ~ ( ord_less_rat @ Y2 @ X2 )
     => ( ( ~ ( ord_less_rat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_3157_antisym__conv3,axiom,
    ! [Y2: num,X2: num] :
      ( ~ ( ord_less_num @ Y2 @ X2 )
     => ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_3158_antisym__conv3,axiom,
    ! [Y2: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y2 @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_3159_antisym__conv3,axiom,
    ! [Y2: int,X2: int] :
      ( ~ ( ord_less_int @ Y2 @ X2 )
     => ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv3
thf(fact_3160_linorder__cases,axiom,
    ! [X2: real,Y2: real] :
      ( ~ ( ord_less_real @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_real @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_3161_linorder__cases,axiom,
    ! [X2: rat,Y2: rat] :
      ( ~ ( ord_less_rat @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_rat @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_3162_linorder__cases,axiom,
    ! [X2: num,Y2: num] :
      ( ~ ( ord_less_num @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_num @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_3163_linorder__cases,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_3164_linorder__cases,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_int @ X2 @ Y2 )
     => ( ( X2 != Y2 )
       => ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_cases
thf(fact_3165_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_3166_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_3167_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_3168_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_3169_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_3170_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_3171_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_3172_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_3173_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_3174_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_3175_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X7: nat] : ( P2 @ X7 ) )
    = ( ^ [P3: nat > $o] :
        ? [N: nat] :
          ( ( P3 @ N )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N )
             => ~ ( P3 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_3176_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: real] : ( P @ A3 @ A3 )
       => ( ! [A3: real,B3: real] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_3177_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: rat] : ( P @ A3 @ A3 )
       => ( ! [A3: rat,B3: rat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_3178_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: num] : ( P @ A3 @ A3 )
       => ( ! [A3: num,B3: num] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_3179_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_3180_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_3181_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3182_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3183_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3184_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3185_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3186_not__less__iff__gr__or__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
      = ( ( ord_less_real @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3187_not__less__iff__gr__or__eq,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ~ ( ord_less_rat @ X2 @ Y2 ) )
      = ( ( ord_less_rat @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3188_not__less__iff__gr__or__eq,axiom,
    ! [X2: num,Y2: num] :
      ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
      = ( ( ord_less_num @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3189_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3190_not__less__iff__gr__or__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
      = ( ( ord_less_int @ Y2 @ X2 )
        | ( X2 = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3191_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3192_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3193_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3194_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3195_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3196_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_3197_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_3198_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_3199_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_3200_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_3201_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3202_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3203_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3204_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3205_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3206_linorder__neqE,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_3207_linorder__neqE,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_rat @ X2 @ Y2 )
       => ( ord_less_rat @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_3208_linorder__neqE,axiom,
    ! [X2: num,Y2: num] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_num @ X2 @ Y2 )
       => ( ord_less_num @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_3209_linorder__neqE,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_nat @ X2 @ Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_3210_linorder__neqE,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_int @ X2 @ Y2 )
       => ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_3211_order__less__asym,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ~ ( ord_less_real @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_3212_order__less__asym,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ~ ( ord_less_rat @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_3213_order__less__asym,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ~ ( ord_less_num @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_3214_order__less__asym,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_3215_order__less__asym,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ~ ( ord_less_int @ Y2 @ X2 ) ) ).

% order_less_asym
thf(fact_3216_linorder__neq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2 != Y2 )
      = ( ( ord_less_real @ X2 @ Y2 )
        | ( ord_less_real @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_3217_linorder__neq__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( X2 != Y2 )
      = ( ( ord_less_rat @ X2 @ Y2 )
        | ( ord_less_rat @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_3218_linorder__neq__iff,axiom,
    ! [X2: num,Y2: num] :
      ( ( X2 != Y2 )
      = ( ( ord_less_num @ X2 @ Y2 )
        | ( ord_less_num @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_3219_linorder__neq__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
      = ( ( ord_less_nat @ X2 @ Y2 )
        | ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_3220_linorder__neq__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 != Y2 )
      = ( ( ord_less_int @ X2 @ Y2 )
        | ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_3221_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_3222_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_3223_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_3224_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_3225_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_3226_order__less__trans,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ( ord_less_real @ Y2 @ Z )
       => ( ord_less_real @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_3227_order__less__trans,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ( ord_less_rat @ Y2 @ Z )
       => ( ord_less_rat @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_3228_order__less__trans,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( ( ord_less_num @ Y2 @ Z )
       => ( ord_less_num @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_3229_order__less__trans,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_3230_order__less__trans,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ord_less_int @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_3231_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3232_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3233_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3234_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3235_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3236_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3237_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3238_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3239_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3240_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3241_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3242_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3243_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3244_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3245_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3246_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3247_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3248_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3249_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3250_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3251_order__less__irrefl,axiom,
    ! [X2: real] :
      ~ ( ord_less_real @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_3252_order__less__irrefl,axiom,
    ! [X2: rat] :
      ~ ( ord_less_rat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_3253_order__less__irrefl,axiom,
    ! [X2: num] :
      ~ ( ord_less_num @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_3254_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_3255_order__less__irrefl,axiom,
    ! [X2: int] :
      ~ ( ord_less_int @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_3256_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3257_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3258_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_num @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3259_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_nat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3260_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_int @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3261_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3262_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3263_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_num @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3264_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_nat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3265_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_int @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3266_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3267_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3268_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3269_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3270_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3271_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3272_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3273_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3274_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3275_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3276_order__less__not__sym,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ~ ( ord_less_real @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_3277_order__less__not__sym,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ~ ( ord_less_rat @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_3278_order__less__not__sym,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ~ ( ord_less_num @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_3279_order__less__not__sym,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_3280_order__less__not__sym,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ~ ( ord_less_int @ Y2 @ X2 ) ) ).

% order_less_not_sym
thf(fact_3281_order__less__imp__triv,axiom,
    ! [X2: real,Y2: real,P: $o] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3282_order__less__imp__triv,axiom,
    ! [X2: rat,Y2: rat,P: $o] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ( ord_less_rat @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3283_order__less__imp__triv,axiom,
    ! [X2: num,Y2: num,P: $o] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( ( ord_less_num @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3284_order__less__imp__triv,axiom,
    ! [X2: nat,Y2: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3285_order__less__imp__triv,axiom,
    ! [X2: int,Y2: int,P: $o] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_int @ Y2 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3286_linorder__less__linear,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_real @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_3287_linorder__less__linear,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_rat @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_3288_linorder__less__linear,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_num @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_3289_linorder__less__linear,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_nat @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_3290_linorder__less__linear,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
      | ( X2 = Y2 )
      | ( ord_less_int @ Y2 @ X2 ) ) ).

% linorder_less_linear
thf(fact_3291_order__less__imp__not__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_3292_order__less__imp__not__eq,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_3293_order__less__imp__not__eq,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_3294_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_3295_order__less__imp__not__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( X2 != Y2 ) ) ).

% order_less_imp_not_eq
thf(fact_3296_order__less__imp__not__eq2,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_3297_order__less__imp__not__eq2,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_3298_order__less__imp__not__eq2,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_3299_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_3300_order__less__imp__not__eq2,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( Y2 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_3301_order__less__imp__not__less,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ~ ( ord_less_real @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_3302_order__less__imp__not__less,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ~ ( ord_less_rat @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_3303_order__less__imp__not__less,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ~ ( ord_less_num @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_3304_order__less__imp__not__less,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ~ ( ord_less_nat @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_3305_order__less__imp__not__less,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ~ ( ord_less_int @ Y2 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_3306_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_set_nat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_3307_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C4 )
       => ( ord_less_set_nat @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_3308_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_3309_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ord_less_set_nat @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_3310_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% psubset_imp_subset
thf(fact_3311_Collect__mono__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
      = ( ! [X: product_prod_nat_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_3312_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X: complex] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_3313_Collect__mono__iff,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
      = ( ! [X: real] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_3314_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_3315_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_3316_set__eq__subset,axiom,
    ( ( ^ [Y4: set_nat,Z2: set_nat] : Y4 = Z2 )
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_3317_subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_3318_Collect__mono,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3319_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_3320_Collect__mono,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_3321_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3322_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3323_subset__refl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% subset_refl
thf(fact_3324_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A5 )
         => ( member_real @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_3325_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A5 )
         => ( member_int @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_3326_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A5 )
         => ( member_complex @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_3327_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
        ! [T2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T2 @ A5 )
         => ( member8440522571783428010at_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_3328_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_3329_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_3330_equalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% equalityD2
thf(fact_3331_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% equalityD1
thf(fact_3332_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A5 )
         => ( member_real @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_3333_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A5 )
         => ( member_int @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_3334_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
        ! [X: complex] :
          ( ( member_complex @ X @ A5 )
         => ( member_complex @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_3335_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_3336_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A5 )
         => ( member_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_3337_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).

% equalityE
thf(fact_3338_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).

% psubsetE
thf(fact_3339_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3340_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3341_subsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3342_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3343_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3344_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real @ X2 @ A2 )
       => ( member_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_3345_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X2: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ X2 @ A2 )
       => ( member_int @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_3346_in__mono,axiom,
    ! [A2: set_complex,B2: set_complex,X2: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( member_complex @ X2 @ A2 )
       => ( member_complex @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_3347_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( member8440522571783428010at_nat @ X2 @ A2 )
       => ( member8440522571783428010at_nat @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_3348_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ X2 @ A2 )
       => ( member_nat @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_3349_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3350_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_3351_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D4: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D4 @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3352_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_3353_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_3354_real__arch__pow__inv,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N4 ) @ Y2 ) ) ) ).

% real_arch_pow_inv
thf(fact_3355_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3356_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_3357_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 )
            & ! [Y3: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y3 )
                  & ( ( power_power_real @ Y3 @ N2 )
                    = A ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3358_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_3359_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_3360_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A5 )
          @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3361_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A5 )
          @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3362_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X: complex] : ( member_complex @ X @ A5 )
          @ ^ [X: complex] : ( member_complex @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3363_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3364_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A5 )
          @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3365_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3366_Collect__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3367_Collect__subset,axiom,
    ! [A2: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X: complex] :
            ( ( member_complex @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3368_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3369_Collect__subset,axiom,
    ! [A2: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3370_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3371_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_3372_leD,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ Y2 @ X2 )
     => ~ ( ord_less_real @ X2 @ Y2 ) ) ).

% leD
thf(fact_3373_leD,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ X2 )
     => ~ ( ord_less_set_nat @ X2 @ Y2 ) ) ).

% leD
thf(fact_3374_leD,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y2 @ X2 )
     => ~ ( ord_less_rat @ X2 @ Y2 ) ) ).

% leD
thf(fact_3375_leD,axiom,
    ! [Y2: num,X2: num] :
      ( ( ord_less_eq_num @ Y2 @ X2 )
     => ~ ( ord_less_num @ X2 @ Y2 ) ) ).

% leD
thf(fact_3376_leD,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y2 ) ) ).

% leD
thf(fact_3377_leD,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ Y2 @ X2 )
     => ~ ( ord_less_int @ X2 @ Y2 ) ) ).

% leD
thf(fact_3378_leI,axiom,
    ! [X2: real,Y2: real] :
      ( ~ ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_eq_real @ Y2 @ X2 ) ) ).

% leI
thf(fact_3379_leI,axiom,
    ! [X2: rat,Y2: rat] :
      ( ~ ( ord_less_rat @ X2 @ Y2 )
     => ( ord_less_eq_rat @ Y2 @ X2 ) ) ).

% leI
thf(fact_3380_leI,axiom,
    ! [X2: num,Y2: num] :
      ( ~ ( ord_less_num @ X2 @ Y2 )
     => ( ord_less_eq_num @ Y2 @ X2 ) ) ).

% leI
thf(fact_3381_leI,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% leI
thf(fact_3382_leI,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_int @ X2 @ Y2 )
     => ( ord_less_eq_int @ Y2 @ X2 ) ) ).

% leI
thf(fact_3383_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_3384_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_3385_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_3386_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_3387_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_3388_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_3389_antisym__conv1,axiom,
    ! [X2: real,Y2: real] :
      ( ~ ( ord_less_real @ X2 @ Y2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_3390_antisym__conv1,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ~ ( ord_less_set_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_set_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_3391_antisym__conv1,axiom,
    ! [X2: rat,Y2: rat] :
      ( ~ ( ord_less_rat @ X2 @ Y2 )
     => ( ( ord_less_eq_rat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_3392_antisym__conv1,axiom,
    ! [X2: num,Y2: num] :
      ( ~ ( ord_less_num @ X2 @ Y2 )
     => ( ( ord_less_eq_num @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_3393_antisym__conv1,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_3394_antisym__conv1,axiom,
    ! [X2: int,Y2: int] :
      ( ~ ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_eq_int @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv1
thf(fact_3395_antisym__conv2,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_3396_antisym__conv2,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ~ ( ord_less_set_nat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_3397_antisym__conv2,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ( ~ ( ord_less_rat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_3398_antisym__conv2,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
     => ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_3399_antisym__conv2,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_3400_antisym__conv2,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% antisym_conv2
thf(fact_3401_dense__ge,axiom,
    ! [Z: real,Y2: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z @ X3 )
         => ( ord_less_eq_real @ Y2 @ X3 ) )
     => ( ord_less_eq_real @ Y2 @ Z ) ) ).

% dense_ge
thf(fact_3402_dense__ge,axiom,
    ! [Z: rat,Y2: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z @ X3 )
         => ( ord_less_eq_rat @ Y2 @ X3 ) )
     => ( ord_less_eq_rat @ Y2 @ Z ) ) ).

% dense_ge
thf(fact_3403_dense__le,axiom,
    ! [Y2: real,Z: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y2 )
         => ( ord_less_eq_real @ X3 @ Z ) )
     => ( ord_less_eq_real @ Y2 @ Z ) ) ).

% dense_le
thf(fact_3404_dense__le,axiom,
    ! [Y2: rat,Z: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
         => ( ord_less_eq_rat @ X3 @ Z ) )
     => ( ord_less_eq_rat @ Y2 @ Z ) ) ).

% dense_le
thf(fact_3405_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3406_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ~ ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3407_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3408_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3409_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3410_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3411_not__le__imp__less,axiom,
    ! [Y2: real,X2: real] :
      ( ~ ( ord_less_eq_real @ Y2 @ X2 )
     => ( ord_less_real @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_3412_not__le__imp__less,axiom,
    ! [Y2: rat,X2: rat] :
      ( ~ ( ord_less_eq_rat @ Y2 @ X2 )
     => ( ord_less_rat @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_3413_not__le__imp__less,axiom,
    ! [Y2: num,X2: num] :
      ( ~ ( ord_less_eq_num @ Y2 @ X2 )
     => ( ord_less_num @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_3414_not__le__imp__less,axiom,
    ! [Y2: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ord_less_nat @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_3415_not__le__imp__less,axiom,
    ! [Y2: int,X2: int] :
      ( ~ ( ord_less_eq_int @ Y2 @ X2 )
     => ( ord_less_int @ X2 @ Y2 ) ) ).

% not_le_imp_less
thf(fact_3416_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_real @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3417_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_set_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3418_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3419_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3420_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3421_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3422_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3423_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3424_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3425_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3426_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3427_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3428_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3429_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3430_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3431_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3432_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3433_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3434_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3435_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3436_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3437_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3438_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3439_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3440_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3441_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3442_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3443_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3444_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3445_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3446_dense__ge__bounded,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z @ W2 )
           => ( ( ord_less_real @ W2 @ X2 )
             => ( ord_less_eq_real @ Y2 @ W2 ) ) )
       => ( ord_less_eq_real @ Y2 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_3447_dense__ge__bounded,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ Z @ W2 )
           => ( ( ord_less_rat @ W2 @ X2 )
             => ( ord_less_eq_rat @ Y2 @ W2 ) ) )
       => ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_3448_dense__le__bounded,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X2 @ W2 )
           => ( ( ord_less_real @ W2 @ Y2 )
             => ( ord_less_eq_real @ W2 @ Z ) ) )
       => ( ord_less_eq_real @ Y2 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_3449_dense__le__bounded,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ X2 @ W2 )
           => ( ( ord_less_rat @ W2 @ Y2 )
             => ( ord_less_eq_rat @ W2 @ Z ) ) )
       => ( ord_less_eq_rat @ Y2 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_3450_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_real @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3451_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( ord_less_set_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3452_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_rat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3453_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_num @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3454_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3455_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_int @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3456_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3457_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3458_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3459_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3460_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3461_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3462_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_3463_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_3464_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_3465_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_3466_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_3467_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_3468_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_3469_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_3470_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_3471_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_3472_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_3473_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_3474_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_3475_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_3476_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_3477_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_3478_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_3479_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_3480_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_3481_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_3482_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_3483_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_3484_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_3485_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_3486_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_3487_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_3488_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_3489_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_3490_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_3491_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_3492_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_3493_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_set_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_3494_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_3495_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_num @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_3496_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_3497_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_3498_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_3499_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( ord_less_eq_set_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_3500_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_3501_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_3502_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_3503_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_3504_linorder__not__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ~ ( ord_less_eq_real @ X2 @ Y2 ) )
      = ( ord_less_real @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_3505_linorder__not__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ~ ( ord_less_eq_rat @ X2 @ Y2 ) )
      = ( ord_less_rat @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_3506_linorder__not__le,axiom,
    ! [X2: num,Y2: num] :
      ( ( ~ ( ord_less_eq_num @ X2 @ Y2 ) )
      = ( ord_less_num @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_3507_linorder__not__le,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
      = ( ord_less_nat @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_3508_linorder__not__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
      = ( ord_less_int @ Y2 @ X2 ) ) ).

% linorder_not_le
thf(fact_3509_linorder__not__less,axiom,
    ! [X2: real,Y2: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
      = ( ord_less_eq_real @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_3510_linorder__not__less,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ~ ( ord_less_rat @ X2 @ Y2 ) )
      = ( ord_less_eq_rat @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_3511_linorder__not__less,axiom,
    ! [X2: num,Y2: num] :
      ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
      = ( ord_less_eq_num @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_3512_linorder__not__less,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
      = ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_3513_linorder__not__less,axiom,
    ! [X2: int,Y2: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
      = ( ord_less_eq_int @ Y2 @ X2 ) ) ).

% linorder_not_less
thf(fact_3514_order__less__imp__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_3515_order__less__imp__le,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_set_nat @ X2 @ Y2 )
     => ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_3516_order__less__imp__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_3517_order__less__imp__le,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( ord_less_eq_num @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_3518_order__less__imp__le,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_3519_order__less__imp__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% order_less_imp_le
thf(fact_3520_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_3521_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_3522_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( A != B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_3523_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_3524_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_3525_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_3526_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_3527_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_3528_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_3529_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_3530_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_3531_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_3532_order__le__less__trans,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ( ord_less_real @ Y2 @ Z )
       => ( ord_less_real @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_3533_order__le__less__trans,axiom,
    ! [X2: set_nat,Y2: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ord_less_set_nat @ Y2 @ Z )
       => ( ord_less_set_nat @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_3534_order__le__less__trans,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ( ord_less_rat @ Y2 @ Z )
       => ( ord_less_rat @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_3535_order__le__less__trans,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
     => ( ( ord_less_num @ Y2 @ Z )
       => ( ord_less_num @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_3536_order__le__less__trans,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ Y2 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_3537_order__le__less__trans,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ord_less_int @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_3538_order__less__le__trans,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ Z )
       => ( ord_less_real @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_3539_order__less__le__trans,axiom,
    ! [X2: set_nat,Y2: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_set_nat @ Y2 @ Z )
       => ( ord_less_set_nat @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_3540_order__less__le__trans,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ( ord_less_eq_rat @ Y2 @ Z )
       => ( ord_less_rat @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_3541_order__less__le__trans,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ord_less_num @ X2 @ Y2 )
     => ( ( ord_less_eq_num @ Y2 @ Z )
       => ( ord_less_num @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_3542_order__less__le__trans,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_3543_order__less__le__trans,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ord_less_int @ X2 @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z )
       => ( ord_less_int @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_3544_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3545_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3546_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_num @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3547_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_nat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3548_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_int @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3549_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3550_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3551_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_num @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3552_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_nat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3553_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_int @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_3554_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3555_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3556_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3557_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3558_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3559_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3560_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3561_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3562_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3563_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_3564_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3565_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3566_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3567_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3568_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3569_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3570_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3571_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3572_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3573_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_eq_num @ X3 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_3574_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3575_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3576_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_num @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3577_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_nat @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3578_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_int @ X3 @ Y5 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3579_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y5: real] :
              ( ( ord_less_real @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3580_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y5: rat] :
              ( ( ord_less_rat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3581_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y5: num] :
              ( ( ord_less_num @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3582_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y5: nat] :
              ( ( ord_less_nat @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3583_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: int,Y5: int] :
              ( ( ord_less_int @ X3 @ Y5 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_3584_linorder__le__less__linear,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
      | ( ord_less_real @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_3585_linorder__le__less__linear,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
      | ( ord_less_rat @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_3586_linorder__le__less__linear,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
      | ( ord_less_num @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_3587_linorder__le__less__linear,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
      | ( ord_less_nat @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_3588_linorder__le__less__linear,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
      | ( ord_less_int @ Y2 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_3589_order__le__imp__less__or__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ( ord_less_real @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_3590_order__le__imp__less__or__eq,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ord_less_set_nat @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_3591_order__le__imp__less__or__eq,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ( ord_less_rat @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_3592_order__le__imp__less__or__eq,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
     => ( ( ord_less_num @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_3593_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_nat @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_3594_order__le__imp__less__or__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_less_int @ X2 @ Y2 )
        | ( X2 = Y2 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_3595_bot_Oextremum,axiom,
    ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).

% bot.extremum
thf(fact_3596_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_3597_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_3598_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_3599_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_3600_bot_Oextremum__unique,axiom,
    ! [A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
      = ( A = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_unique
thf(fact_3601_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_3602_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_3603_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_3604_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_3605_bot_Oextremum__uniqueI,axiom,
    ! [A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
     => ( A = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_3606_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_3607_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_3608_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_3609_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_3610_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_3611_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).

% bot.extremum_strict
thf(fact_3612_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_3613_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_3614_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_3615_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_3616_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
      = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_3617_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_3618_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_3619_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_3620_max__def,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A4: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3621_max__def,axiom,
    ( ord_max_Code_integer
    = ( ^ [A4: code_integer,B4: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3622_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3623_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3624_max__def,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3625_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3626_max__def,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_3627_max__absorb1,axiom,
    ! [Y2: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y2 @ X2 )
     => ( ( ord_ma741700101516333627d_enat @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3628_max__absorb1,axiom,
    ! [Y2: code_integer,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y2 @ X2 )
     => ( ( ord_max_Code_integer @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3629_max__absorb1,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ X2 )
     => ( ( ord_max_set_nat @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3630_max__absorb1,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y2 @ X2 )
     => ( ( ord_max_rat @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3631_max__absorb1,axiom,
    ! [Y2: num,X2: num] :
      ( ( ord_less_eq_num @ Y2 @ X2 )
     => ( ( ord_max_num @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3632_max__absorb1,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ( ord_max_nat @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3633_max__absorb1,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ Y2 @ X2 )
     => ( ( ord_max_int @ X2 @ Y2 )
        = X2 ) ) ).

% max_absorb1
thf(fact_3634_max__absorb2,axiom,
    ! [X2: extended_enat,Y2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
     => ( ( ord_ma741700101516333627d_enat @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3635_max__absorb2,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X2 @ Y2 )
     => ( ( ord_max_Code_integer @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3636_max__absorb2,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ord_max_set_nat @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3637_max__absorb2,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ( ord_max_rat @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3638_max__absorb2,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_num @ X2 @ Y2 )
     => ( ( ord_max_num @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3639_max__absorb2,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_max_nat @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3640_max__absorb2,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ( ord_max_int @ X2 @ Y2 )
        = Y2 ) ) ).

% max_absorb2
thf(fact_3641_vebt__pred_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y2
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y2 = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va2 ) ) )
                     => ( ( ( B3
                           => ( Y2
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B3
                           => ( ( A3
                               => ( Y2
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y2 = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
                     => ( ( Y2 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                       => ( ( Y2 = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y2
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y2
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_3642_vebt__delete_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y2
                    = ( vEBT_Leaf @ $false @ B3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2
                      = ( vEBT_Leaf @ A3 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N4: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2
                          = ( vEBT_Leaf @ A3 @ B3 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                     => ( ( Y2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                       => ( ( Y2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y2
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y2
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y2
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va2 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va2 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_3643_vebt__insert_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y2
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y2
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y2
                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
               => ( ( Y2
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
                 => ( ( Y2
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y2
                        = ( 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,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y2
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_3644_vebt__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y2
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( 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] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y2
                   => ~ ( 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] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y2
                     => ~ ( 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,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y2
                          = ( ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3645_vebt__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( 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] :
                  ( ( X2
                    = ( 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] :
                    ( ( X2
                      = ( 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,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3646_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( 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] :
                  ( ( X2
                    = ( 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_3647_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X2
                  = ( 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_3648_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y2
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( 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] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( Y2
                      = ( ( ( 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_3649_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_3650_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_3651_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_3652_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_3653_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_3654_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_3655_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_3656_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_3657_vebt__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_3658_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_3659_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_3660_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y2
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y2
                      = ( ( 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 @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y2
                        = ( ( 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] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( Y2
                          = ( ( ( 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_3661_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( 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,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( 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] :
                      ( ( X2
                        = ( 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_3662_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X2
                  = ( 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] :
                  ( ( X2
                    = ( 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_3663_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_3664_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_3665_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_3666_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_3667_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_3668_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_3669_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_3670_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_3671_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_3672_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_3673_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_3674_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_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_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_3683_Icc__eq__Icc,axiom,
    ! [L2: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3684_Icc__eq__Icc,axiom,
    ! [L2: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3685_Icc__eq__Icc,axiom,
    ! [L2: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L2 @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3686_Icc__eq__Icc,axiom,
    ! [L2: nat,H2: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3687_Icc__eq__Icc,axiom,
    ! [L2: int,H2: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L2 @ H2 )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3688_Icc__eq__Icc,axiom,
    ! [L2: real,H2: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L2 @ H2 )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3689_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L2: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
      = ( ( ord_less_eq_set_nat @ L2 @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3690_atLeastAtMost__iff,axiom,
    ! [I: rat,L2: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
      = ( ( ord_less_eq_rat @ L2 @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3691_atLeastAtMost__iff,axiom,
    ! [I: num,L2: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
      = ( ( ord_less_eq_num @ L2 @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3692_atLeastAtMost__iff,axiom,
    ! [I: nat,L2: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3693_atLeastAtMost__iff,axiom,
    ! [I: int,L2: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( ( ord_less_eq_int @ L2 @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3694_atLeastAtMost__iff,axiom,
    ! [I: real,L2: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
      = ( ( ord_less_eq_real @ L2 @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3695_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_3696_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_3697_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_3698_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_3699_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_3700_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_3701_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M7: nat] :
      ( ( P @ X2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M7 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3702_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc3368934014287244435at_num] :
      ~ ! [F2: nat > num > num,A3: nat,B3: nat,Acc: num] :
          ( X2
         != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A3 @ ( product_Pair_nat_num @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_3703_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X2
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_3704_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3705_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ K @ I4 )
             => ( ( P @ I4 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3706_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_3707_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I4: int] :
              ( ( ord_less_int @ K @ I4 )
             => ( ( P @ I4 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3708_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_3709_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I4: int] :
              ( ( ord_less_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_3710_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_3711_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_3712_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_3713_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_3714_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_3715_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_3716_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_3717_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_3718_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3719_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_3720_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_3721_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I4: int] :
            ( ( ord_less_eq_int @ K @ I4 )
           => ( ( P @ I4 )
             => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3722_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_3723_cppi,axiom,
    ! [D4: int,P: int > $o,P5: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( P @ X3 )
              = ( P5 @ 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,K2: int] :
                ( ( P5 @ X3 )
                = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P5 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A2 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3724_cpmi,axiom,
    ! [D4: int,P: int > $o,P5: int > $o,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( P @ X3 )
              = ( P5 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B2 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P5 @ X3 )
                = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P5 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B2 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3725_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B2 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_3726_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X4 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_3727_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_3728_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X4 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_3729_bset_I3_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 = T )
             => ( ( minus_minus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_3730_bset_I4_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 != T )
             => ( ( minus_minus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_3731_pinf_I1_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3732_pinf_I1_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3733_pinf_I1_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3734_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3735_pinf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3736_pinf_I2_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3737_pinf_I2_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3738_pinf_I2_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3739_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3740_pinf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z3 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3741_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3742_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3743_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3744_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3745_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3746_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3747_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3748_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3749_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3750_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3751_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ~ ( ord_less_real @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3752_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ~ ( ord_less_rat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3753_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ~ ( ord_less_num @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3754_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3755_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ~ ( ord_less_int @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3756_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ord_less_real @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3757_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ord_less_rat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3758_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( ord_less_num @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3759_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ord_less_nat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3760_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ord_less_int @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3761_minf_I1_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3762_minf_I1_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3763_minf_I1_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3764_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3765_minf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3766_minf_I2_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3767_minf_I2_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3768_minf_I2_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3769_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3770_minf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3771_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3772_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3773_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3774_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3775_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3776_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3777_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3778_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3779_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3780_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3781_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ord_less_real @ X4 @ T ) ) ).

% minf(5)
thf(fact_3782_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ord_less_rat @ X4 @ T ) ) ).

% minf(5)
thf(fact_3783_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( ord_less_num @ X4 @ T ) ) ).

% minf(5)
thf(fact_3784_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ord_less_nat @ X4 @ T ) ) ).

% minf(5)
thf(fact_3785_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ord_less_int @ X4 @ T ) ) ).

% minf(5)
thf(fact_3786_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ~ ( ord_less_real @ T @ X4 ) ) ).

% minf(7)
thf(fact_3787_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ~ ( ord_less_rat @ T @ X4 ) ) ).

% minf(7)
thf(fact_3788_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ~ ( ord_less_num @ T @ X4 ) ) ).

% minf(7)
thf(fact_3789_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ~ ( ord_less_nat @ T @ X4 ) ) ).

% minf(7)
thf(fact_3790_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ~ ( ord_less_int @ T @ X4 ) ) ).

% minf(7)
thf(fact_3791_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ~ ( ord_less_eq_real @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3792_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3793_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ~ ( ord_less_eq_num @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3794_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3795_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3796_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ord_less_eq_real @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3797_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ord_less_eq_rat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3798_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( ord_less_eq_num @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3799_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3800_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ord_less_eq_int @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3801_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ord_less_eq_real @ X4 @ T ) ) ).

% minf(6)
thf(fact_3802_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ord_less_eq_rat @ X4 @ T ) ) ).

% minf(6)
thf(fact_3803_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( ord_less_eq_num @ X4 @ T ) ) ).

% minf(6)
thf(fact_3804_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_3805_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ord_less_eq_int @ X4 @ T ) ) ).

% minf(6)
thf(fact_3806_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ~ ( ord_less_eq_real @ T @ X4 ) ) ).

% minf(8)
thf(fact_3807_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).

% minf(8)
thf(fact_3808_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ~ ( ord_less_eq_num @ T @ X4 ) ) ).

% minf(8)
thf(fact_3809_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_3810_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ~ ( ord_less_eq_int @ T @ X4 ) ) ).

% minf(8)
thf(fact_3811_inf__period_I1_J,axiom,
    ! [P: complex > $o,D4: complex,Q: complex > $o] :
      ( ! [X3: complex,K2: complex] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
     => ( ! [X3: complex,K2: complex] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
       => ! [X4: complex,K4: complex] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3812_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X4: real,K4: real] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3813_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X4: int,K4: int] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3814_inf__period_I2_J,axiom,
    ! [P: complex > $o,D4: complex,Q: complex > $o] :
      ( ! [X3: complex,K2: complex] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
     => ( ! [X3: complex,K2: complex] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K2 @ D4 ) ) ) )
       => ! [X4: complex,K4: complex] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3815_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X4: real,K4: real] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3816_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X4: int,K4: int] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3817_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [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] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_3818_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [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] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_3819_bset_I2_J,axiom,
    ! [D4: int,B2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_3820_bset_I1_J,axiom,
    ! [D4: int,B2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_3821_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X5: int] : ( P @ X5 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3822_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X4 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_3823_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X4 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_3824_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 != T )
             => ( ( plus_plus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_3825_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 = T )
             => ( ( plus_plus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_3826_bset_I7_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X4 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_3827_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B2 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X4 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_3828_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_3829_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_3830_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A3: real,B3: real,C3: real] :
            ( ( P @ A3 @ B3 )
           => ( ( P @ B3 @ C3 )
             => ( ( ord_less_eq_real @ A3 @ B3 )
               => ( ( ord_less_eq_real @ B3 @ C3 )
                 => ( P @ A3 @ C3 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A3: real,B3: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B3 )
                          & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D5 ) )
                       => ( P @ A3 @ B3 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_3831_mult__le__cancel__iff2,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y2 ) )
        = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3832_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y2 ) )
        = ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3833_mult__le__cancel__iff2,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y2 ) )
        = ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3834_mult__le__cancel__iff1,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ Z ) )
        = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3835_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ Z ) )
        = ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3836_mult__le__cancel__iff1,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y2 @ Z ) )
        = ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3837_divides__aux__eq,axiom,
    ! [Q2: nat,R: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
      = ( R = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_3838_divides__aux__eq,axiom,
    ! [Q2: int,R: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
      = ( R = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_3839_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
       => ( 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 @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_3840_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_3841_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_3842_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_3843_mult__less__iff1,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ Z ) )
        = ( ord_less_real @ X2 @ Y2 ) ) ) ).

% mult_less_iff1
thf(fact_3844_mult__less__iff1,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ Z ) )
        = ( ord_less_rat @ X2 @ Y2 ) ) ) ).

% mult_less_iff1
thf(fact_3845_mult__less__iff1,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y2 @ Z ) )
        = ( ord_less_int @ X2 @ Y2 ) ) ) ).

% mult_less_iff1
thf(fact_3846_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
       => ( 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 @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_3847_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% low_def
thf(fact_3848_obtain__set__pred,axiom,
    ! [Z: nat,X2: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X2 @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_3849_obtain__set__succ,axiom,
    ! [X2: nat,Z: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B2 )
         => ( ( A2 = B2 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X2 @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_3850_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_3851_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 )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs2 )
              & ( ord_less_nat @ A @ X4 ) ) ) ) ).

% succ_none_empty
thf(fact_3852_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 )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs2 )
              & ( ord_less_nat @ X4 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_3853_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_3854_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_3855_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_3856_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_3857_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_3858_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: $o,A6: code_integer,B6: $o] :
      ( ( ( produc6677183202524767010eger_o @ A @ B )
        = ( produc6677183202524767010eger_o @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_3859_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A6: num,B6: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_3860_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A6: nat,B6: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_3861_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A6: nat,B6: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_3862_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A6: int,B6: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_3863_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_3864_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_3865_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_3866_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_3867_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_3868_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_3869_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_3870_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_3871_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_3872_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3873_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3874_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3875_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3876_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_3877_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_3878_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_3879_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_3880_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_3881_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_3882_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_3883_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_3884_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_3885_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_3886_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_3887_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_3888_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_3889_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_3890_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_3891_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_3892_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_3893_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_3894_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_3895_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_3896_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_3897_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_3898_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_3899_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_3900_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_3901_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_3902_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_3903_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_3904_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_3905_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_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_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_3913_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_3914_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_3915_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_3916_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_3917_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_3918_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_3919_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_3920_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_3921_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_3922_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_3923_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_3924_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_3925_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A6 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B6 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_3926_mod__mult__cong,axiom,
    ! [A: int,C: int,A6: int,B: int,B6: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A6 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B6 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_3927_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_3928_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_3929_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_3930_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_3931_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_3932_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_3933_mod__add__cong,axiom,
    ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A6 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B6 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3934_mod__add__cong,axiom,
    ! [A: int,C: int,A6: int,B: int,B6: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A6 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B6 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3935_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_3936_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_3937_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_3938_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_3939_mod__diff__cong,axiom,
    ! [A: int,C: int,A6: int,B: int,B6: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A6 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B6 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_3940_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_3941_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_3942_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_3943_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_3944_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_3945_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_3946_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N6 )
         => ( ord_less_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_3947_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N3 )
         => ( ord_less_nat @ X3 @ N2 ) )
     => ( finite_finite_nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_3948_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N6 )
         => ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_3949_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3950_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3951_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3952_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3953_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3954_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3955_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_3956_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_3957_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_3958_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_3959_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_3960_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_3961_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_3962_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_3963_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_3964_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_3965_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_3966_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_3967_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_3968_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_3969_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_3970_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_3971_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_3972_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_3973_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_3974_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P6: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P6 )
       => ( ( ord_less_nat @ M @ P6 )
         => ( ! [N4: nat] :
                ( ( ord_less_nat @ N4 @ P6 )
               => ( ( P @ N4 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P6 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_3975_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_3976_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_3977_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_3978_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_3979_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_3980_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_3981_nat__mod__eq__iff,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ( modulo_modulo_nat @ X2 @ N2 )
        = ( modulo_modulo_nat @ Y2 @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X2 @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y2 @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3982_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3983_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3984_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_3985_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_3986_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_3987_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_3988_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_3989_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_3990_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_3991_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_3992_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_3993_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3994_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3995_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_3996_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_3997_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_3998_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_3999_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4000_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4001_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_4002_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_4003_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_4004_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_4005_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_4006_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_4007_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_4008_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_4009_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_4010_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_4011_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_4012_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_4013_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_4014_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_4015_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_4016_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_4017_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_4018_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_4019_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_4020_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_4021_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_4022_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_4023_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_4024_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_4025_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_4026_nat__mod__eq__lemma,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ( modulo_modulo_nat @ X2 @ N2 )
        = ( modulo_modulo_nat @ Y2 @ N2 ) )
     => ( ( ord_less_eq_nat @ Y2 @ X2 )
       => ? [Q3: nat] :
            ( X2
            = ( plus_plus_nat @ Y2 @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_4027_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S2: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_4028_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_4029_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_4030_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N ) @ N ) ) ) ) ).

% modulo_nat_def
thf(fact_4031_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc6271795597528267376eger_o] :
      ~ ! [A3: code_integer,B3: $o] :
          ( Y2
         != ( produc6677183202524767010eger_o @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_4032_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_num_num] :
      ~ ! [A3: num,B3: num] :
          ( Y2
         != ( product_Pair_num_num @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_4033_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_num] :
      ~ ! [A3: nat,B3: num] :
          ( Y2
         != ( product_Pair_nat_num @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_4034_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y2
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_4035_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y2
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_4036_surj__pair,axiom,
    ! [P6: produc6271795597528267376eger_o] :
    ? [X3: code_integer,Y5: $o] :
      ( P6
      = ( produc6677183202524767010eger_o @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_4037_surj__pair,axiom,
    ! [P6: product_prod_num_num] :
    ? [X3: num,Y5: num] :
      ( P6
      = ( product_Pair_num_num @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_4038_surj__pair,axiom,
    ! [P6: product_prod_nat_num] :
    ? [X3: nat,Y5: num] :
      ( P6
      = ( product_Pair_nat_num @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_4039_surj__pair,axiom,
    ! [P6: product_prod_nat_nat] :
    ? [X3: nat,Y5: nat] :
      ( P6
      = ( product_Pair_nat_nat @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_4040_surj__pair,axiom,
    ! [P6: product_prod_int_int] :
    ? [X3: int,Y5: int] :
      ( P6
      = ( product_Pair_int_int @ X3 @ Y5 ) ) ).

% surj_pair
thf(fact_4041_prod__cases,axiom,
    ! [P: produc6271795597528267376eger_o > $o,P6: produc6271795597528267376eger_o] :
      ( ! [A3: code_integer,B3: $o] : ( P @ ( produc6677183202524767010eger_o @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_4042_prod__cases,axiom,
    ! [P: product_prod_num_num > $o,P6: product_prod_num_num] :
      ( ! [A3: num,B3: num] : ( P @ ( product_Pair_num_num @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_4043_prod__cases,axiom,
    ! [P: product_prod_nat_num > $o,P6: product_prod_nat_num] :
      ( ! [A3: nat,B3: num] : ( P @ ( product_Pair_nat_num @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_4044_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P6: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_4045_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P6: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_4046_Pair__inject,axiom,
    ! [A: code_integer,B: $o,A6: code_integer,B6: $o] :
      ( ( ( produc6677183202524767010eger_o @ A @ B )
        = ( produc6677183202524767010eger_o @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B = ~ B6 ) ) ) ).

% Pair_inject
thf(fact_4047_Pair__inject,axiom,
    ! [A: num,B: num,A6: num,B6: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_4048_Pair__inject,axiom,
    ! [A: nat,B: num,A6: nat,B6: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_4049_Pair__inject,axiom,
    ! [A: nat,B: nat,A6: nat,B6: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_4050_Pair__inject,axiom,
    ! [A: int,B: int,A6: int,B6: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_4051_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_4052_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_4053_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_4054_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_4055_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_4056_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_4057_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_4058_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_4059_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_4060_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_4061_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_4062_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_4063_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_4064_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_4065_mod__double__modulus,axiom,
    ! [M: nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
       => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X2 @ M ) )
          | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4066_mod__double__modulus,axiom,
    ! [M: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X2 @ M ) )
          | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4067_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_4068_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_4069_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_4070_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_4071_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_4072_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_4073_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_4074_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_4075_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B2 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_4076_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_4077_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_4078_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_4079_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_4080_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z5: real] :
              ( ( power_power_real @ Z5 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_4081_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_4082_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_4083_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_4084_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_4085_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B2 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_4086_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_4087_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_4088_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_4089_mod__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_4090_mod__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_4091_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_4092_neg__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% neg_mod_bound
thf(fact_4093_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_4094_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N4: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X4 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_4095_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N4: nat] :
        ! [X4: list_o] :
          ( ( member_list_o @ X4 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_4096_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N4: nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_4097_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N4: nat] :
        ! [X4: list_int] :
          ( ( member_list_int @ X4 @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_4098_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_4099_neg__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_4100_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4101_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4102_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4103_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4104_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4105_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_4106_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_4107_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_4108_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_4109_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4110_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4111_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4112_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4113_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4114_mod__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( plus_plus_int @ K @ L2 ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_4115_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B2 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_4116_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_4117_verit__comp__simplify1_I3_J,axiom,
    ! [B6: real,A6: real] :
      ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
      = ( ord_less_real @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4118_verit__comp__simplify1_I3_J,axiom,
    ! [B6: rat,A6: rat] :
      ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
      = ( ord_less_rat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4119_verit__comp__simplify1_I3_J,axiom,
    ! [B6: num,A6: num] :
      ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
      = ( ord_less_num @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4120_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
      = ( ord_less_nat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4121_verit__comp__simplify1_I3_J,axiom,
    ! [B6: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
      = ( ord_less_int @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4122_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_4123_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_4124_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_4125_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_4126_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_4127_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_4128_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_4129_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_4130_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_4131_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_4132_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_4133_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_4134_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_4135_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_4136_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_4137_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_4138_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_4139_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_4140_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_4141_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_4142_infinite__super,axiom,
    ! [S: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_4143_infinite__super,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T3 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_4144_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_4145_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_4146_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_4147_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A4: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4148_max__def__raw,axiom,
    ( ord_max_Code_integer
    = ( ^ [A4: code_integer,B4: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4149_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4150_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4151_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4152_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4153_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_4154_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_4155_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_4156_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_4157_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_4158_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_4159_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_4160_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_4161_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_4162_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_4163_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_4164_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_4165_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_4166_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_4167_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_4168_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_4169_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_4170_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_4171_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_4172_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_4173_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_4174_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N2 )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ 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_4175_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N2 )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ 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_4176_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_o_o @ ( nth_o @ 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_4177_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_o_nat @ ( nth_o @ 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_4178_product__nth,axiom,
    ! [N2: nat,Xs2: list_o,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_o_int @ ( nth_o @ 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_4179_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: int > rat,Y2: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_rat @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4180_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: real > rat,Y2: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_rat @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4181_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: nat > rat,Y2: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( times_times_rat @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4182_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: complex > rat,Y2: complex > rat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( times_times_rat @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4183_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: int > complex,Y2: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4184_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: real > complex,Y2: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4185_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: nat > complex,Y2: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( times_times_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4186_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: complex > complex,Y2: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( times_times_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4187_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: int > real,Y2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_real @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4188_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: real > real,Y2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_real @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_4189_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: int > complex,Y2: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4190_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: real > complex,Y2: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4191_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: nat > complex,Y2: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4192_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: complex > complex,Y2: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4193_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: int > real,Y2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4194_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: real > real,Y2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4195_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X2: nat > real,Y2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4196_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X2: complex > real,Y2: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4197_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X2: int > rat,Y2: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_rat @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4198_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X2: real > rat,Y2: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X2 @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y2 @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_rat @ ( X2 @ I3 ) @ ( Y2 @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4199_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_4200_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_4201_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_4202_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_4203_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_4204_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_4205_length__product,axiom,
    ! [Xs2: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_4206_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_4207_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_4208_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_4209_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
     => ( ! [M5: nat,N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ( P @ N4 @ ( modulo_modulo_nat @ M5 @ N4 ) )
             => ( P @ M5 @ N4 ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_4210_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L2 )
      = ( 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 ) ) ) @ L2 ) ) ) ) ).

% concat_bit_Suc
thf(fact_4211_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4212_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4213_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4214_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4215_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_4216_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_4217_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_4218_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_4219_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_4220_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_4221_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_4222_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_4223_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_4224_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_4225_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_4226_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_4227_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_4228_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_4229_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_4230_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_4231_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_4232_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_4233_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_4234_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_4235_concat__bit__0,axiom,
    ! [K: int,L2: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
      = L2 ) ).

% concat_bit_0
thf(fact_4236_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_4237_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_4238_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_4239_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_4240_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_4241_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_4242_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_4243_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_4244_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_4245_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_4246_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_4247_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_4248_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_4249_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_4250_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_4251_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_4252_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_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_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_4260_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_4261_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_4262_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_4263_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_4264_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_4265_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_4266_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_4267_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_4268_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_4269_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_4270_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_4271_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_4272_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_4273_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_4274_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_4275_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_4276_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_4277_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_4278_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_4279_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_4280_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_4281_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_4282_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_4283_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_4284_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_4285_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_4286_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_4287_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_4288_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_4289_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_4290_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_4291_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).

% concat_bit_nonnegative_iff
thf(fact_4292_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L2 ) @ zero_zero_int )
      = ( ord_less_int @ L2 @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_4293_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_4294_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_4295_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_4296_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_4297_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_4298_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_4299_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_4300_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_4301_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_4302_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_4303_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_4304_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_4305_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_4306_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_4307_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_4308_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_4309_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_4310_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_4311_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_4312_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_4313_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_4314_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_4315_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_4316_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_4317_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_4318_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_4319_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_4320_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_4321_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_4322_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_4323_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_4324_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_4325_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_4326_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_4327_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_4328_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_4329_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_4330_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_4331_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_4332_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_4333_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_4334_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_4335_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_4336_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_4337_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_4338_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_4339_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_4340_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_4341_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_4342_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_4343_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_4344_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_4345_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_4346_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_4347_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_4348_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_4349_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_4350_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_4351_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_4352_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_4353_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_4354_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_4355_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_4356_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_4357_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_4358_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_4359_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_4360_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_4361_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_4362_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_4363_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_4364_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_4365_dvd__productE,axiom,
    ! [P6: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y5: nat] :
            ( ( P6
              = ( times_times_nat @ X3 @ Y5 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y5 @ B ) ) ) ) ).

% dvd_productE
thf(fact_4366_dvd__productE,axiom,
    ! [P6: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y5: int] :
            ( ( P6
              = ( times_times_int @ X3 @ Y5 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y5 @ B ) ) ) ) ).

% dvd_productE
thf(fact_4367_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B7: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B7 @ C5 ) )
          & ( dvd_dvd_nat @ B7 @ B )
          & ( dvd_dvd_nat @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_4368_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B7: int,C5: int] :
          ( ( A
            = ( times_times_int @ B7 @ C5 ) )
          & ( dvd_dvd_int @ B7 @ B )
          & ( dvd_dvd_int @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_4369_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).

% dvdE
thf(fact_4370_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ~ ! [K2: complex] :
            ( A
           != ( times_times_complex @ B @ K2 ) ) ) ).

% dvdE
thf(fact_4371_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B @ K2 ) ) ) ).

% dvdE
thf(fact_4372_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_4373_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_4374_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_4375_dvdI,axiom,
    ! [A: complex,B: complex,K: complex] :
      ( ( A
        = ( times_times_complex @ B @ K ) )
     => ( dvd_dvd_complex @ B @ A ) ) ).

% dvdI
thf(fact_4376_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_4377_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_4378_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_4379_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B4: code_integer,A4: code_integer] :
        ? [K3: code_integer] :
          ( A4
          = ( times_3573771949741848930nteger @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_4380_dvd__def,axiom,
    ( dvd_dvd_complex
    = ( ^ [B4: complex,A4: complex] :
        ? [K3: complex] :
          ( A4
          = ( times_times_complex @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_4381_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B4: real,A4: real] :
        ? [K3: real] :
          ( A4
          = ( times_times_real @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_4382_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B4: nat,A4: nat] :
        ? [K3: nat] :
          ( A4
          = ( times_times_nat @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_4383_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B4: int,A4: int] :
        ? [K3: int] :
          ( A4
          = ( times_times_int @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_4384_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_4385_dvd__mult,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ C )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4386_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_4387_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_4388_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_4389_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_4390_dvd__mult2,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4391_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_4392_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_4393_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_4394_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_4395_dvd__mult__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
     => ( dvd_dvd_complex @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4396_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_4397_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_4398_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_4399_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4400_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4401_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4402_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4403_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4404_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_4405_mult__dvd__mono,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( ( dvd_dvd_complex @ C @ D )
       => ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4406_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_4407_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_4408_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_4409_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_4410_dvd__mult__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
     => ( dvd_dvd_complex @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4411_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_4412_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_4413_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_4414_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4415_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4416_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4417_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4418_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4419_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_4420_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_4421_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_4422_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_4423_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_4424_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_4425_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_4426_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_4427_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_4428_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_4429_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_4430_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_4431_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_4432_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_4433_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_4434_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_4435_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_4436_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_4437_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_4438_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_4439_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_4440_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_4441_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_4442_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_4443_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_4444_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_4445_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_4446_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_4447_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_4448_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_4449_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_4450_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_4451_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_4452_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_4453_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_4454_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_4455_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_4456_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_4457_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_4458_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_4459_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_4460_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_4461_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_4462_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_4463_dvd__power__same,axiom,
    ! [X2: code_integer,Y2: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ X2 @ Y2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N2 ) @ ( power_8256067586552552935nteger @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4464_dvd__power__same,axiom,
    ! [X2: nat,Y2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ X2 @ Y2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4465_dvd__power__same,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( dvd_dvd_real @ X2 @ Y2 )
     => ( dvd_dvd_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4466_dvd__power__same,axiom,
    ! [X2: int,Y2: int,N2: nat] :
      ( ( dvd_dvd_int @ X2 @ Y2 )
     => ( dvd_dvd_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4467_dvd__power__same,axiom,
    ! [X2: complex,Y2: complex,N2: nat] :
      ( ( dvd_dvd_complex @ X2 @ Y2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_4468_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_4469_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_4470_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_4471_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_4472_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_4473_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_4474_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_4475_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_4476_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_4477_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X2: nat,Y2: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X2 )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
            | ( ( times_times_nat @ B @ X2 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D ) ) )
         => ? [X3: nat,Y5: 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 ) @ Y5 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_4478_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y5 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y5 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_4479_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4480_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4481_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4482_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4483_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4484_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L2: int,R: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L2 ) @ R ) ) ).

% concat_bit_assoc
thf(fact_4485_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_4486_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_4487_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_4488_minf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_4489_minf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_4490_minf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_4491_minf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_4492_minf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) ) ) ) ) ).

% minf(10)
thf(fact_4493_minf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_4494_minf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_4495_minf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_4496_minf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_4497_minf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) ) ) ) ).

% minf(9)
thf(fact_4498_pinf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_4499_pinf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_4500_pinf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_4501_pinf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_4502_pinf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) ) ) ) ) ).

% pinf(10)
thf(fact_4503_pinf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_4504_pinf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_4505_pinf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_4506_pinf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_4507_pinf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S3 ) ) ) ) ).

% pinf(9)
thf(fact_4508_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_4509_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_4510_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_4511_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_4512_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_4513_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_4514_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_4515_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_4516_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_4517_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_4518_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_4519_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_4520_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_4521_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_4522_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_4523_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_4524_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_4525_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_4526_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_4527_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_4528_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_4529_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_4530_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_4531_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_4532_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_4533_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_4534_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_4535_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_4536_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_4537_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_4538_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_4539_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_4540_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_4541_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_4542_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_4543_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_4544_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_4545_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_4546_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_4547_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_4548_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_4549_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_4550_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_4551_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_4552_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_4553_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_4554_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_4555_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_4556_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_4557_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_4558_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_4559_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_4560_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_4561_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_4562_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_4563_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_4564_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_4565_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_4566_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_4567_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_4568_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_4569_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_4570_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_4571_dvd__power__le,axiom,
    ! [X2: code_integer,Y2: code_integer,N2: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N2 ) @ ( power_8256067586552552935nteger @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4572_dvd__power__le,axiom,
    ! [X2: nat,Y2: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4573_dvd__power__le,axiom,
    ! [X2: real,Y2: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4574_dvd__power__le,axiom,
    ! [X2: int,Y2: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4575_dvd__power__le,axiom,
    ! [X2: complex,Y2: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4576_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_4577_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_4578_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_4579_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_4580_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_4581_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_4582_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_4583_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_4584_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_4585_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_4586_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_4587_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_4588_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y5: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_4589_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_4590_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_4591_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_4592_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_4593_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_4594_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).

% dbl_def
thf(fact_4595_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).

% dbl_def
thf(fact_4596_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).

% dbl_def
thf(fact_4597_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_4598_div2__even__ext__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y2 ) )
       => ( X2 = Y2 ) ) ) ).

% div2_even_ext_nat
thf(fact_4599_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L2: code_integer] :
      ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X ) ) )
      = ( ? [X: code_integer] :
            ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4600_unity__coeff__ex,axiom,
    ! [P: rat > $o,L2: rat] :
      ( ( ? [X: rat] : ( P @ ( times_times_rat @ L2 @ X ) ) )
      = ( ? [X: rat] :
            ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X @ zero_zero_rat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4601_unity__coeff__ex,axiom,
    ! [P: complex > $o,L2: complex] :
      ( ( ? [X: complex] : ( P @ ( times_times_complex @ L2 @ X ) ) )
      = ( ? [X: complex] :
            ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X @ zero_zero_complex ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4602_unity__coeff__ex,axiom,
    ! [P: real > $o,L2: real] :
      ( ( ? [X: real] : ( P @ ( times_times_real @ L2 @ X ) ) )
      = ( ? [X: real] :
            ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X @ zero_zero_real ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4603_unity__coeff__ex,axiom,
    ! [P: nat > $o,L2: nat] :
      ( ( ? [X: nat] : ( P @ ( times_times_nat @ L2 @ X ) ) )
      = ( ? [X: nat] :
            ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X @ zero_zero_nat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4604_unity__coeff__ex,axiom,
    ! [P: int > $o,L2: int] :
      ( ( ? [X: int] : ( P @ ( times_times_int @ L2 @ X ) ) )
      = ( ? [X: int] :
            ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X @ zero_zero_int ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4605_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4606_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4607_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4608_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_4609_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_4610_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_4611_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_4612_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_4613_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_4614_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_4615_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_4616_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_4617_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_4618_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_4619_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_4620_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_4621_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_4622_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_4623_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_4624_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_4625_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_4626_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X4: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4627_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X4: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4628_inf__period_I4_J,axiom,
    ! [D: complex,D4: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X4: complex,K4: complex] :
          ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4629_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X4: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4630_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4631_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X4: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4632_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X4: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4633_inf__period_I3_J,axiom,
    ! [D: complex,D4: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X4: complex,K4: complex] :
          ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X4 @ T ) )
          = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X4 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4634_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X4: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X4 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4635_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4636_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_4637_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_4638_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_4639_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_4640_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_4641_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_4642_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_4643_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_4644_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_4645_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_4646_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_4647_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_4648_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_4649_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_4650_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_4651_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_4652_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_4653_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_4654_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_4655_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_4656_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_4657_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_4658_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_4659_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_4660_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_4661_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_4662_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_4663_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_4664_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N2 @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4665_prod__decode__aux_Ocases,axiom,
    ! [X2: product_prod_nat_nat] :
      ~ ! [K2: nat,M5: nat] :
          ( X2
         != ( product_Pair_nat_nat @ K2 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_4666_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_4667_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_4668_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_4669_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4670_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4671_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4672_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B3 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B3 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4673_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B3 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B3 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4674_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( ( dvd_dvd_int @ B3 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B3 )
                 => ( ( ( divide_divide_int @ one_one_int @ B3 )
                      = A )
                   => ( ( ( times_times_int @ A @ B3 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4675_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_4676_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_4677_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_4678_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_4679_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_4680_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_4681_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_4682_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_4683_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_4684_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4685_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4686_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4687_bit__eq__rec,axiom,
    ( ( ^ [Y4: code_integer,Z2: code_integer] : Y4 = Z2 )
    = ( ^ [A4: code_integer,B4: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4688_bit__eq__rec,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : Y4 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4689_bit__eq__rec,axiom,
    ( ( ^ [Y4: int,Z2: int] : Y4 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4690_dvd__power__iff,axiom,
    ! [X2: code_integer,M: nat,N2: nat] :
      ( ( X2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ M ) @ ( power_8256067586552552935nteger @ X2 @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X2 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4691_dvd__power__iff,axiom,
    ! [X2: nat,M: nat,N2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X2 @ M ) @ ( power_power_nat @ X2 @ N2 ) )
        = ( ( dvd_dvd_nat @ X2 @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4692_dvd__power__iff,axiom,
    ! [X2: int,M: nat,N2: nat] :
      ( ( X2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ N2 ) )
        = ( ( dvd_dvd_int @ X2 @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4693_dvd__power,axiom,
    ! [N2: nat,X2: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X2 @ ( power_8256067586552552935nteger @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_4694_dvd__power,axiom,
    ! [N2: nat,X2: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_rat ) )
     => ( dvd_dvd_rat @ X2 @ ( power_power_rat @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_4695_dvd__power,axiom,
    ! [N2: nat,X2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_nat ) )
     => ( dvd_dvd_nat @ X2 @ ( power_power_nat @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_4696_dvd__power,axiom,
    ! [N2: nat,X2: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_real ) )
     => ( dvd_dvd_real @ X2 @ ( power_power_real @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_4697_dvd__power,axiom,
    ! [N2: nat,X2: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_int ) )
     => ( dvd_dvd_int @ X2 @ ( power_power_int @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_4698_dvd__power,axiom,
    ! [N2: nat,X2: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_complex ) )
     => ( dvd_dvd_complex @ X2 @ ( power_power_complex @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_4699_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_4700_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_4701_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_4702_dvd__minus__add,axiom,
    ! [Q2: nat,N2: nat,R: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N2 )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4703_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_4704_mod__nat__eqI,axiom,
    ! [R: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R @ N2 )
     => ( ( ord_less_eq_nat @ R @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R ) ) ) ) ).

% mod_nat_eqI
thf(fact_4705_bset_I9_J,axiom,
    ! [D: int,D4: int,B2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B2 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_4706_bset_I10_J,axiom,
    ! [D: int,D4: int,B2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B2 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_4707_aset_I9_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_4708_aset_I10_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_4709_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_4710_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_4711_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_4712_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_4713_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_4714_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_4715_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_4716_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_4717_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_4718_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_4719_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_4720_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_4721_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_4722_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_4723_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_4724_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_4725_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_4726_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_4727_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_4728_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_4729_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_4730_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_4731_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_4732_even__diff__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_diff_iff
thf(fact_4733_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4734_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_4735_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_4736_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_4737_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_4738_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_4739_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_4740_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_4741_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_4742_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_4743_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_4744_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_4745_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_4746_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_4747_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_4748_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_4749_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_4750_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_4751_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N4: nat] :
            ( X2
           != ( suc @ N4 ) ) ) ).

% list_decode.cases
thf(fact_4752_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_4753_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_4754_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_4755_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_4756_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_4757_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_4758_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_4759_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_4760_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_4761_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_4762_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_4763_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_4764_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_4765_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_4766_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_4767_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_4768_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_4769_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_4770_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_4771_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_4772_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_4773_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_4774_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S )
                  & ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4775_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( ( finite_finite_rat @ S )
     => ( ( S != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S )
                  & ( ord_less_rat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4776_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( ( finite_finite_num @ S )
     => ( ( S != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S )
                  & ( ord_less_num @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4777_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S )
                  & ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4778_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S )
                  & ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4779_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_4780_vebt__buildup_Oelims,axiom,
    ! [X2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_4781_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_4782_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_4783_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_4784_option_Osize__gen_I2_J,axiom,
    ! [X2: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X2 @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4785_option_Osize__gen_I2_J,axiom,
    ! [X2: nat > nat,X22: nat] :
      ( ( size_option_nat @ X2 @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4786_option_Osize__gen_I2_J,axiom,
    ! [X2: num > nat,X22: num] :
      ( ( size_option_num @ X2 @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4787_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_4788_set__decode__Suc,axiom,
    ! [N2: nat,X2: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X2 ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_4789_diff__shunt__var,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( ( minus_minus_set_int @ X2 @ Y2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_4790_diff__shunt__var,axiom,
    ! [X2: set_real,Y2: set_real] :
      ( ( ( minus_minus_set_real @ X2 @ Y2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_4791_diff__shunt__var,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ( minus_minus_set_nat @ X2 @ Y2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_4792_intind,axiom,
    ! [I: nat,N2: nat,P: nat > $o,X2: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_4793_intind,axiom,
    ! [I: nat,N2: nat,P: int > $o,X2: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_int @ ( replicate_int @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_4794_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_4795_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_4796_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_4797_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_4798_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_4799_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_4800_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_4801_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_4802_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_4803_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_4804_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_4805_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_4806_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4807_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4808_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4809_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_4810_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4811_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4812_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4813_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4814_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4815_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4816_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_4817_replicate__eq__replicate,axiom,
    ! [M: nat,X2: vEBT_VEBT,N2: nat,Y2: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X2 )
        = ( replicate_VEBT_VEBT @ N2 @ Y2 ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X2 = Y2 ) ) ) ) ).

% replicate_eq_replicate
thf(fact_4818_length__replicate,axiom,
    ! [N2: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_4819_length__replicate,axiom,
    ! [N2: nat,X2: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_4820_length__replicate,axiom,
    ! [N2: nat,X2: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_4821_length__replicate,axiom,
    ! [N2: nat,X2: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_4822_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_4823_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_4824_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_4825_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_4826_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_4827_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_4828_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_4829_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4830_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_4831_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_4832_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_4833_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_4834_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_4835_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_4836_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_4837_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_4838_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_4839_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_4840_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_4841_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_4842_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_4843_in__set__replicate,axiom,
    ! [X2: real,N2: nat,Y2: real] :
      ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4844_in__set__replicate,axiom,
    ! [X2: complex,N2: nat,Y2: complex] :
      ( ( member_complex @ X2 @ ( set_complex2 @ ( replicate_complex @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4845_in__set__replicate,axiom,
    ! [X2: product_prod_nat_nat,N2: nat,Y2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4846_in__set__replicate,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4847_in__set__replicate,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4848_in__set__replicate,axiom,
    ! [X2: vEBT_VEBT,N2: nat,Y2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4849_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ? [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4850_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4851_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4852_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4853_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4854_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4855_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_4856_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_4857_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_4858_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_4859_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_4860_odd__of__bool__self,axiom,
    ! [P6: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P6 ) ) )
      = P6 ) ).

% odd_of_bool_self
thf(fact_4861_odd__of__bool__self,axiom,
    ! [P6: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P6 ) ) )
      = P6 ) ).

% odd_of_bool_self
thf(fact_4862_odd__of__bool__self,axiom,
    ! [P6: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P6 ) ) )
      = P6 ) ).

% odd_of_bool_self
thf(fact_4863_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_4864_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_4865_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_4866_set__decode__0,axiom,
    ! [X2: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).

% set_decode_0
thf(fact_4867_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_4868_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_4869_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_4870_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_4871_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_4872_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_4873_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_4874_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_4875_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_4876_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_4877_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n1201886186963655149omplex
        @ ( P
          & Q ) )
      = ( times_times_complex @ ( zero_n1201886186963655149omplex @ P ) @ ( zero_n1201886186963655149omplex @ Q ) ) ) ).

% of_bool_conj
thf(fact_4878_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_4879_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_4880_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_4881_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_4882_signed__take__bit__mult,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).

% signed_take_bit_mult
thf(fact_4883_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L2 ) ) ) ).

% signed_take_bit_add
thf(fact_4884_signed__take__bit__diff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% signed_take_bit_diff
thf(fact_4885_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_4886_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_4887_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_4888_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_4889_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4890_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_4891_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_4892_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_4893_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_4894_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_4895_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_4896_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_4897_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_4898_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_4899_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_4900_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_4901_split__of__bool,axiom,
    ! [P: complex > $o,P6: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
      = ( ( P6
         => ( P @ one_one_complex ) )
        & ( ~ P6
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_4902_split__of__bool,axiom,
    ! [P: real > $o,P6: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
      = ( ( P6
         => ( P @ one_one_real ) )
        & ( ~ P6
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_4903_split__of__bool,axiom,
    ! [P: rat > $o,P6: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
      = ( ( P6
         => ( P @ one_one_rat ) )
        & ( ~ P6
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_4904_split__of__bool,axiom,
    ! [P: nat > $o,P6: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
      = ( ( P6
         => ( P @ one_one_nat ) )
        & ( ~ P6
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_4905_split__of__bool,axiom,
    ! [P: int > $o,P6: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
      = ( ( P6
         => ( P @ one_one_int ) )
        & ( ~ P6
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_4906_split__of__bool,axiom,
    ! [P: code_integer > $o,P6: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
      = ( ( P6
         => ( P @ one_one_Code_integer ) )
        & ( ~ P6
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_4907_split__of__bool__asm,axiom,
    ! [P: complex > $o,P6: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
      = ( ~ ( ( P6
              & ~ ( P @ one_one_complex ) )
            | ( ~ P6
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4908_split__of__bool__asm,axiom,
    ! [P: real > $o,P6: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
      = ( ~ ( ( P6
              & ~ ( P @ one_one_real ) )
            | ( ~ P6
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4909_split__of__bool__asm,axiom,
    ! [P: rat > $o,P6: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
      = ( ~ ( ( P6
              & ~ ( P @ one_one_rat ) )
            | ( ~ P6
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4910_split__of__bool__asm,axiom,
    ! [P: nat > $o,P6: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
      = ( ~ ( ( P6
              & ~ ( P @ one_one_nat ) )
            | ( ~ P6
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4911_split__of__bool__asm,axiom,
    ! [P: int > $o,P6: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
      = ( ~ ( ( P6
              & ~ ( P @ one_one_int ) )
            | ( ~ P6
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4912_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P6: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
      = ( ~ ( ( P6
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P6
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4913_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4914_replicate__length__same,axiom,
    ! [Xs2: list_o,X2: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4915_replicate__length__same,axiom,
    ! [Xs2: list_nat,X2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4916_replicate__length__same,axiom,
    ! [Xs2: list_int,X2: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4917_replicate__eqI,axiom,
    ! [Xs2: list_real,N2: nat,X2: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N2 )
     => ( ! [Y5: real] :
            ( ( member_real @ Y5 @ ( set_real2 @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replicate_real @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4918_replicate__eqI,axiom,
    ! [Xs2: list_complex,N2: nat,X2: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N2 )
     => ( ! [Y5: complex] :
            ( ( member_complex @ Y5 @ ( set_complex2 @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replicate_complex @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4919_replicate__eqI,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,N2: nat,X2: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs2 )
        = N2 )
     => ( ! [Y5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replic4235873036481779905at_nat @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4920_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N2: nat,X2: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N2 )
     => ( ! [Y5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4921_replicate__eqI,axiom,
    ! [Xs2: list_o,N2: nat,X2: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N2 )
     => ( ! [Y5: $o] :
            ( ( member_o @ Y5 @ ( set_o2 @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replicate_o @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4922_replicate__eqI,axiom,
    ! [Xs2: list_nat,N2: nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N2 )
     => ( ! [Y5: nat] :
            ( ( member_nat @ Y5 @ ( set_nat2 @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replicate_nat @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4923_replicate__eqI,axiom,
    ! [Xs2: list_int,N2: nat,X2: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N2 )
     => ( ! [Y5: int] :
            ( ( member_int @ Y5 @ ( set_int2 @ Xs2 ) )
           => ( Y5 = X2 ) )
       => ( Xs2
          = ( replicate_int @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_4924_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_4925_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_4926_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_4927_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_4928_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_4929_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_4930_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_4931_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A3: nat] :
          ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: nat,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4932_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A3: int] :
          ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: int,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4933_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: code_integer,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4934_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_4935_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_4936_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_4937_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_4938_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_4939_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_4940_option_Osize__gen_I1_J,axiom,
    ! [X2: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X2 @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4941_option_Osize__gen_I1_J,axiom,
    ! [X2: nat > nat] :
      ( ( size_option_nat @ X2 @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4942_option_Osize__gen_I1_J,axiom,
    ! [X2: num > nat] :
      ( ( size_option_num @ X2 @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4943_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect_nat
          @ ^ [N: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% set_decode_def
thf(fact_4944_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_4945_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_4946_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_4947_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_4948_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N: nat,A4: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4949_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A4: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4950_vebt__buildup_Opelims,axiom,
    ! [X2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y2
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y2
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_4951_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4952_add__scale__eq__noteq,axiom,
    ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4953_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4954_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4955_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4956_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_4957_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( 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_4958_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4959_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4960_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4961_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_4962_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) )
      = ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4963_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_4964_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_4965_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_4966_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_4967_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_4968_compl__le__compl__iff,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y2 ) )
      = ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ).

% compl_le_compl_iff
thf(fact_4969_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_4970_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_4971_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_4972_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_4973_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_4974_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_4975_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4976_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4977_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_4978_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_4979_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_4980_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_4981_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_4982_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_4983_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_4984_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_4985_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_4986_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_4987_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_4988_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_4989_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_4990_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_4991_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_4992_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_4993_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_4994_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_4995_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4996_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4997_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_4998_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_4999_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_5000_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5001_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5002_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_5003_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_5004_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_5005_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_5006_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_5007_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_5008_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_5009_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_5010_ln__inj__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ( ln_ln_real @ X2 )
            = ( ln_ln_real @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% ln_inj_iff
thf(fact_5011_ln__less__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
          = ( ord_less_real @ X2 @ Y2 ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5012_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_5013_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_5014_real__add__minus__iff,axiom,
    ! [X2: real,A: real] :
      ( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X2 = A ) ) ).

% real_add_minus_iff
thf(fact_5015_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5016_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5017_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_5018_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_5019_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_5020_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_5021_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5022_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_5023_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_5024_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_5025_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_5026_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_5027_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_5028_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_5029_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_5030_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_5031_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_5032_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_5033_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_5034_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_5035_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_5036_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_5037_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_5038_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_5039_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_5040_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_5041_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_5042_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_5043_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_5044_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_5045_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_5046_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_5047_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_5048_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_5049_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_5050_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_5051_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_5052_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_5053_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_5054_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_5055_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_5056_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_5057_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_5058_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_5059_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_5060_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_5061_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_5062_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_5063_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_5064_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_5065_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_5066_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_5067_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_5068_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_5069_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_5070_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_5071_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_5072_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_5073_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_5074_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_5075_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_5076_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_5077_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_5078_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_5079_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_5080_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_5081_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_5082_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_5083_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_5084_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_5085_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_5086_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_5087_divide__minus1,axiom,
    ! [X2: real] :
      ( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X2 ) ) ).

% divide_minus1
thf(fact_5088_divide__minus1,axiom,
    ! [X2: complex] :
      ( ( divide1717551699836669952omplex @ X2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X2 ) ) ).

% divide_minus1
thf(fact_5089_divide__minus1,axiom,
    ! [X2: rat] :
      ( ( divide_divide_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X2 ) ) ).

% divide_minus1
thf(fact_5090_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_5091_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_5092_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_5093_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_5094_ln__le__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
          = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_5095_ln__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_5096_ln__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
        = ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% ln_gt_zero_iff
thf(fact_5097_ln__eq__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ln_ln_real @ X2 )
          = zero_zero_real )
        = ( X2 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_5098_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_5099_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_5100_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_5101_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_5102_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_5103_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5104_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5105_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_5106_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_5107_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_5108_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_5109_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_5110_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_5111_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_5112_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_5113_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_5114_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_5115_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_5116_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_5117_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_5118_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_5119_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_5120_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_5121_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_5122_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_5123_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_5124_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_5125_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_5126_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_5127_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_5128_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_5129_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_5130_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_5131_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_5132_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_5133_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_5134_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_5135_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_5136_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_5137_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_5138_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_5139_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_5140_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_5141_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5142_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_5143_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_5144_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_5145_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_5146_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_5147_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_5148_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_5149_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_5150_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_5151_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_5152_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_5153_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_5154_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_5155_ln__ge__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
        = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% ln_ge_zero_iff
thf(fact_5156_ln__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_5157_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_5158_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_5159_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_5160_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_5161_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_5162_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_5163_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_5164_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_5165_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_5166_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_5167_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_5168_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_5169_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_5170_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_5171_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_5172_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_5173_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_5174_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_5175_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_5176_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_5177_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_5178_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_5179_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_5180_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_5181_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_5182_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_5183_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_5184_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_5185_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_5186_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_5187_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_5188_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_5189_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_5190_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y2 ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_5191_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_5192_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_5193_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_5194_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_5195_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y2 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_5196_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_5197_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y2 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_5198_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y2 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_5199_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y2 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_5200_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y2 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_5201_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y2 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_5202_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_5203_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_5204_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_5205_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_5206_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_5207_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_5208_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_5209_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_5210_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_5211_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_5212_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_5213_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_5214_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_5215_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_5216_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_5217_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_5218_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_5219_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_5220_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_5221_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_5222_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_5223_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_5224_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_5225_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_5226_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_5227_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_5228_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_5229_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_5230_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_5231_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_5232_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_5233_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_5234_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_5235_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_5236_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_5237_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_5238_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_5239_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_5240_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_5241_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_5242_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_5243_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_5244_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_5245_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_5246_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_5247_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_5248_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_5249_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_5250_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_5251_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_5252_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_5253_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_5254_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_5255_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_5256_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_5257_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_5258_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_5259_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_5260_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_5261_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_5262_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_5263_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_5264_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_5265_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_5266_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_5267_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_5268_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_5269_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_5270_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_5271_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_5272_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_5273_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_5274_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_5275_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_5276_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_5277_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_5278_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_5279_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_5280_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_5281_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_5282_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_5283_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_5284_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_5285_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_5286_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5287_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5288_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_5289_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_5290_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_5291_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_5292_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_5293_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_5294_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_5295_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_5296_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_5297_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_5298_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_5299_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_5300_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_5301_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_5302_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_5303_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_5304_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_5305_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_5306_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_5307_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_5308_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_5309_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_5310_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_5311_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_5312_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_5313_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_5314_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_5315_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_5316_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_5317_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_5318_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_5319_compl__mono,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ ( uminus5710092332889474511et_nat @ X2 ) ) ) ).

% compl_mono
thf(fact_5320_compl__le__swap1,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ ( uminus5710092332889474511et_nat @ X2 ) )
     => ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).

% compl_le_swap1
thf(fact_5321_compl__le__swap2,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ X2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y2 ) ) ).

% compl_le_swap2
thf(fact_5322_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_5323_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_5324_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_5325_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_5326_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_5327_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_5328_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_5329_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_5330_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_5331_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_5332_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_5333_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_5334_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_5335_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_5336_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5337_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5338_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_5339_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_5340_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_5341_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5342_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5343_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_5344_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_5345_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_5346_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_5347_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_5348_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_5349_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_5350_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_5351_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_5352_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_5353_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_5354_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_5355_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_5356_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_5357_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_5358_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_5359_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_5360_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_5361_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_5362_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_5363_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_5364_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_5365_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_5366_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_5367_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_5368_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_5369_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_5370_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_5371_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_5372_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_5373_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_5374_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_5375_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_5376_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_5377_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_5378_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_5379_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_5380_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_5381_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_5382_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_5383_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5384_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_5385_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_5386_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5387_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_5388_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_5389_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5390_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat,X1: nat,X22: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_5391_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_5392_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_5393_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_5394_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_5395_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_5396_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_5397_mod__minus__cong,axiom,
    ! [A: int,B: int,A6: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A6 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A6 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5398_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A6: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A6 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A6 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5399_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_5400_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5401_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5402_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5403_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5404_sum__mono,axiom,
    ! [K5: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5405_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5406_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5407_sum__mono,axiom,
    ! [K5: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5408_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5409_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5410_sum__mono,axiom,
    ! [K5: set_complex,F: complex > int,G: complex > int] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5411_sum__distrib__left,axiom,
    ! [R: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ R @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5412_sum__distrib__left,axiom,
    ! [R: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( times_times_complex @ R @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5413_sum__distrib__left,axiom,
    ! [R: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ R @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5414_sum__distrib__left,axiom,
    ! [R: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( times_times_real @ R @ ( F @ N ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5415_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5416_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( times_times_complex @ ( F @ N ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5417_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5418_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5419_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B2: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_5420_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B2: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_5421_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_5422_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B2: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_5423_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5424_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5425_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5426_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5427_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( divide1717551699836669952omplex @ ( F @ N ) @ R )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_5428_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ R )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_5429_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_5430_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_5431_case__prodE2,axiom,
    ! [Q: product_prod_nat_nat > $o,P: nat > nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc2626176000494625587at_nat @ P @ Z ) )
     => ~ ! [X3: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y5 ) )
           => ~ ( Q @ ( P @ X3 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_5432_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y5 ) )
           => ~ ( Q @ ( P @ X3 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_5433_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,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y5 ) )
           => ~ ( Q @ ( P @ X3 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_5434_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,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y5 ) )
           => ~ ( Q @ ( P @ X3 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_5435_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,Y5: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y5 ) )
           => ~ ( Q @ ( P @ X3 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_5436_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5437_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5438_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5439_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5440_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5441_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y5: nat] :
          ( ( F @ X3 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
     => ( ( produc2626176000494625587at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5442_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y5: nat] :
          ( ( F @ X3 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5443_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,Y5: nat] :
          ( ( F @ X3 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5444_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,Y5: nat] :
          ( ( F @ X3 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5445_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X3: int,Y5: int] :
          ( ( F @ X3 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y5 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5446_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_5447_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_5448_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_5449_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_5450_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_5451_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_5452_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_5453_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_5454_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_5455_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_5456_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_5457_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_5458_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_5459_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_5460_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_5461_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_5462_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_5463_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_5464_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_5465_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_5466_ln__add__one__self__le__self2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self2
thf(fact_5467_sum__mono__inv,axiom,
    ! [F: real > rat,I6: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I6 )
        = ( groups1300246762558778688al_rat @ G @ I6 ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5468_sum__mono__inv,axiom,
    ! [F: int > rat,I6: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I6 )
        = ( groups3906332499630173760nt_rat @ G @ I6 ) )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_int @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5469_sum__mono__inv,axiom,
    ! [F: nat > rat,I6: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I6 )
        = ( groups2906978787729119204at_rat @ G @ I6 ) )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_nat @ I @ I6 )
         => ( ( finite_finite_nat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5470_sum__mono__inv,axiom,
    ! [F: complex > rat,I6: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I6 )
        = ( groups5058264527183730370ex_rat @ G @ I6 ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5471_sum__mono__inv,axiom,
    ! [F: real > nat,I6: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I6 )
        = ( groups1935376822645274424al_nat @ G @ I6 ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5472_sum__mono__inv,axiom,
    ! [F: int > nat,I6: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
        = ( groups4541462559716669496nt_nat @ G @ I6 ) )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_int @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5473_sum__mono__inv,axiom,
    ! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
        = ( groups5693394587270226106ex_nat @ G @ I6 ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5474_sum__mono__inv,axiom,
    ! [F: real > int,I6: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I6 )
        = ( groups1932886352136224148al_int @ G @ I6 ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5475_sum__mono__inv,axiom,
    ! [F: nat > int,I6: set_nat,G: nat > int,I: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I6 )
        = ( groups3539618377306564664at_int @ G @ I6 ) )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_nat @ I @ I6 )
         => ( ( finite_finite_nat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5476_sum__mono__inv,axiom,
    ! [F: complex > int,I6: set_complex,G: complex > int,I: complex] :
      ( ( ( groups5690904116761175830ex_int @ F @ I6 )
        = ( groups5690904116761175830ex_int @ G @ I6 ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5477_ln__less__self,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).

% ln_less_self
thf(fact_5478_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_5479_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5480_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_5481_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_5482_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_5483_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5484_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_5485_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_5486_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5487_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5488_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5489_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5490_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5491_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_5492_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_5493_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5494_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_5495_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_5496_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_5497_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5498_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_5499_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_5500_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_5501_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_5502_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_5503_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_5504_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_5505_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_5506_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_5507_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_5508_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_5509_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_5510_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_5511_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5512_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5513_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_5514_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_5515_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_5516_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_5517_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_5518_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_5519_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_5520_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_5521_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_5522_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_5523_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_5524_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_5525_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_5526_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_5527_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_5528_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_5529_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_5530_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_5531_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_5532_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5533_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5534_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5535_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5536_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5537_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_5538_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_5539_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_5540_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_5541_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_5542_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_5543_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_5544_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_5545_numeral__times__minus__swap,axiom,
    ! [W: num,X2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X2 ) )
      = ( times_times_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5546_numeral__times__minus__swap,axiom,
    ! [W: num,X2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X2 ) )
      = ( times_times_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5547_numeral__times__minus__swap,axiom,
    ! [W: num,X2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X2 ) )
      = ( times_times_complex @ X2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5548_numeral__times__minus__swap,axiom,
    ! [W: num,X2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X2 ) )
      = ( times_3573771949741848930nteger @ X2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5549_numeral__times__minus__swap,axiom,
    ! [W: num,X2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X2 ) )
      = ( times_times_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5550_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_5551_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_5552_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_5553_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_5554_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_5555_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_5556_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5557_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5558_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5559_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5560_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5561_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ N2 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5562_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ N2 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5563_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ N2 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5564_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ N2 )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5565_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ N2 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5566_square__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( times_times_real @ X2 @ X2 )
        = one_one_real )
      = ( ( X2 = one_one_real )
        | ( X2
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_5567_square__eq__1__iff,axiom,
    ! [X2: int] :
      ( ( ( times_times_int @ X2 @ X2 )
        = one_one_int )
      = ( ( X2 = one_one_int )
        | ( X2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_5568_square__eq__1__iff,axiom,
    ! [X2: complex] :
      ( ( ( times_times_complex @ X2 @ X2 )
        = one_one_complex )
      = ( ( X2 = one_one_complex )
        | ( X2
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_5569_square__eq__1__iff,axiom,
    ! [X2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X2 @ X2 )
        = one_one_Code_integer )
      = ( ( X2 = one_one_Code_integer )
        | ( X2
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5570_square__eq__1__iff,axiom,
    ! [X2: rat] :
      ( ( ( times_times_rat @ X2 @ X2 )
        = one_one_rat )
      = ( ( X2 = one_one_rat )
        | ( X2
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5571_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5572_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5573_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5574_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5575_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5576_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5577_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5578_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5579_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5580_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5581_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5582_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5583_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5584_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5585_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5586_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_5587_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_5588_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_5589_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_5590_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_5591_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_5592_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_5593_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_5594_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_5595_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_5596_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_5597_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_5598_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_5599_real__minus__mult__self__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).

% real_minus_mult_self_le
thf(fact_5600_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_5601_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_5602_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).

% minus_real_def
thf(fact_5603_ln__one__minus__pos__upper__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_5604_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 )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5605_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 )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5606_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 )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5607_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 )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5608_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 )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5609_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 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5610_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 )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5611_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 )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5612_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 )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5613_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 )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5614_sum__le__included,axiom,
    ! [S3: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5615_sum__le__included,axiom,
    ! [S3: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5616_sum__le__included,axiom,
    ! [S3: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5617_sum__le__included,axiom,
    ! [S3: set_complex,T: set_nat,G: nat > rat,I: nat > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5618_sum__le__included,axiom,
    ! [S3: set_complex,T: set_complex,G: complex > rat,I: complex > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5619_sum__le__included,axiom,
    ! [S3: set_complex,T: set_complex,G: complex > nat,I: complex > complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5620_sum__le__included,axiom,
    ! [S3: set_nat,T: set_nat,G: nat > int,I: nat > nat,F: nat > int] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S3 ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5621_sum__le__included,axiom,
    ! [S3: set_nat,T: set_complex,G: complex > int,I: complex > nat,F: nat > int] :
      ( ( finite_finite_nat @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S3 ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5622_sum__le__included,axiom,
    ! [S3: set_complex,T: set_nat,G: nat > int,I: nat > complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S3 ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5623_sum__le__included,axiom,
    ! [S3: set_complex,T: set_complex,G: complex > int,I: complex > complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S3 ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5624_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 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5625_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 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5626_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 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5627_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 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5628_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 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ord_less_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5629_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 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5630_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 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5631_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5632_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5633_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups2073611262835488442omplex @ H2 @ S ) @ ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5634_sum_Orelated,axiom,
    ! [R2: real > real > $o,S: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5635_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S ) @ ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5636_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S ) @ ( groups5058264527183730370ex_rat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5637_sum_Orelated,axiom,
    ! [R2: nat > nat > $o,S: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R2 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5693394587270226106ex_nat @ H2 @ S ) @ ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5638_sum_Orelated,axiom,
    ! [R2: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
      ( ( R2 @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups3539618377306564664at_int @ H2 @ S ) @ ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5639_sum_Orelated,axiom,
    ! [R2: int > int > $o,S: set_complex,H2: complex > int,G: complex > int] :
      ( ( R2 @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups5690904116761175830ex_int @ H2 @ S ) @ ( groups5690904116761175830ex_int @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5640_sum_Orelated,axiom,
    ! [R2: int > int > $o,S: set_int,H2: int > int,G: int > int] :
      ( ( R2 @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups4538972089207619220nt_int @ H2 @ S ) @ ( groups4538972089207619220nt_int @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5641_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups7754918857620584856omplex @ H2 @ S ) @ ( groups7754918857620584856omplex @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5642_sum_Orelated,axiom,
    ! [R2: nat > nat > $o,S: set_nat,H2: nat > nat,G: nat > nat] :
      ( ( R2 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups3542108847815614940at_nat @ H2 @ S ) @ ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_5643_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_5644_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_5645_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_5646_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_5647_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_5648_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_5649_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_5650_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_5651_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_5652_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_5653_ln__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).

% ln_bound
thf(fact_5654_ln__gt__zero__imp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_5655_ln__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_5656_ln__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).

% ln_gt_zero
thf(fact_5657_ln__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).

% ln_ge_zero
thf(fact_5658_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_5659_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5660_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_5661_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_5662_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_5663_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5664_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_5665_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_5666_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_5667_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_5668_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5669_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_5670_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_5671_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_5672_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5673_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_5674_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_5675_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5676_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_5677_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_5678_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_5679_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5680_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_5681_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_5682_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_5683_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_5684_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5685_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_5686_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_5687_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_5688_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5689_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_5690_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_5691_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5692_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_5693_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_5694_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_5695_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5696_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_5697_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_5698_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_5699_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_5700_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_5701_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_5702_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_5703_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5704_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_5705_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_5706_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_5707_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5708_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_5709_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_5710_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_5711_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_5712_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5713_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_5714_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_5715_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_5716_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5717_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_5718_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_5719_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_5720_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5721_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_5722_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_5723_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_5724_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5725_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_5726_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_5727_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_5728_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_5729_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_5730_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_5731_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_5732_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_5733_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_5734_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_5735_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_5736_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_5737_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_5738_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_5739_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_5740_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_5741_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_5742_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_5743_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_5744_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_5745_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_5746_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_5747_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_5748_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_5749_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_5750_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_5751_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_5752_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_5753_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_5754_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_5755_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_5756_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5757_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_5758_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5759_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5760_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_5761_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_5762_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_5763_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_5764_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_5765_power__minus__Bit0,axiom,
    ! [X2: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5766_power__minus__Bit0,axiom,
    ! [X2: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5767_power__minus__Bit0,axiom,
    ! [X2: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5768_power__minus__Bit0,axiom,
    ! [X2: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5769_power__minus__Bit0,axiom,
    ! [X2: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5770_sum__nonneg__0,axiom,
    ! [S3: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_real @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5771_sum__nonneg__0,axiom,
    ! [S3: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_int @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5772_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S3 )
            = zero_zero_real )
         => ( ( member_complex @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5773_sum__nonneg__0,axiom,
    ! [S3: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_real @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5774_sum__nonneg__0,axiom,
    ! [S3: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_int @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5775_sum__nonneg__0,axiom,
    ! [S3: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5776_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5777_sum__nonneg__0,axiom,
    ! [S3: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S3 )
            = zero_zero_nat )
         => ( ( member_real @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5778_sum__nonneg__0,axiom,
    ! [S3: set_int,F: int > nat,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
            = zero_zero_nat )
         => ( ( member_int @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5779_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
            = zero_zero_nat )
         => ( ( member_complex @ I @ S3 )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5780_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: real > real,B2: real,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S3 )
            = B2 )
         => ( ( member_real @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5781_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: int > real,B2: real,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S3 )
            = B2 )
         => ( ( member_int @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5782_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: complex > real,B2: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S3 )
            = B2 )
         => ( ( member_complex @ I @ S3 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5783_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: real > rat,B2: rat,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
            = B2 )
         => ( ( member_real @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5784_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: int > rat,B2: rat,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
            = B2 )
         => ( ( member_int @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5785_sum__nonneg__leq__bound,axiom,
    ! [S3: set_nat,F: nat > rat,B2: rat,I: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
            = B2 )
         => ( ( member_nat @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5786_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: complex > rat,B2: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
            = B2 )
         => ( ( member_complex @ I @ S3 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5787_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: real > nat,B2: nat,I: real] :
      ( ( finite_finite_real @ S3 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S3 )
            = B2 )
         => ( ( member_real @ I @ S3 )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5788_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: int > nat,B2: nat,I: int] :
      ( ( finite_finite_int @ S3 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
            = B2 )
         => ( ( member_int @ I @ S3 )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5789_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: complex > nat,B2: nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
            = B2 )
         => ( ( member_complex @ I @ S3 )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5790_real__add__less__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
      = ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).

% real_add_less_0_iff
thf(fact_5791_real__0__less__add__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).

% real_0_less_add_iff
thf(fact_5792_real__add__le__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).

% real_add_le_0_iff
thf(fact_5793_real__0__le__add__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).

% real_0_le_add_iff
thf(fact_5794_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5795_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5796_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5797_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5798_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5799_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I4: nat] :
                ( ( member_nat @ I4 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5800_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5801_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5802_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5803_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5804_sum__pos,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5805_sum__pos,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5806_sum__pos,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5807_sum__pos,axiom,
    ! [I6: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5808_sum__pos,axiom,
    ! [I6: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5809_sum__pos,axiom,
    ! [I6: set_int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5810_sum__pos,axiom,
    ! [I6: set_real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5811_sum__pos,axiom,
    ! [I6: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5812_sum__pos,axiom,
    ! [I6: set_int,F: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5813_sum__pos,axiom,
    ! [I6: set_real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5814_ln__ge__zero__imp__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_5815_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B2 ) )
                = ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5816_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ A2 )
                  = ( groups3049146728041665814omplex @ H2 @ B2 ) )
                = ( ( groups3049146728041665814omplex @ G @ C4 )
                  = ( groups3049146728041665814omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5817_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5818_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B2 ) )
                = ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5819_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5820_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5821_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B2 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5822_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5823_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5824_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B2 ) )
                = ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5825_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5826_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ C4 )
                  = ( groups3049146728041665814omplex @ H2 @ C4 ) )
               => ( ( groups3049146728041665814omplex @ G @ A2 )
                  = ( groups3049146728041665814omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5827_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5828_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) )
               => ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5829_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5830_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5831_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A2 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5832_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5833_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5834_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) )
               => ( ( groups4541462559716669496nt_nat @ G @ A2 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5835_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5836_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5837_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5838_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5839_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ S )
            = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5840_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5841_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5842_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ S )
            = ( groups4538972089207619220nt_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5843_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ S )
            = ( groups7754918857620584856omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5844_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ S )
            = ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5845_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5846_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5847_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5848_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5849_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ T3 )
            = ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5850_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5851_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5852_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ T3 )
            = ( groups4538972089207619220nt_int @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5853_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ T3 )
            = ( groups7754918857620584856omplex @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5854_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ T3 )
            = ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5855_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5856_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ S )
              = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5857_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5858_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S )
              = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5859_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5860_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5861_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S )
              = ( groups3906332499630173760nt_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5862_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S )
              = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5863_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5864_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ S )
              = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5865_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5866_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ T3 )
              = ( groups3049146728041665814omplex @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5867_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5868_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T3 )
              = ( groups8778361861064173332t_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5869_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5870_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5871_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
              = ( groups3906332499630173760nt_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5872_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5873_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5874_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ T3 )
              = ( groups4541462559716669496nt_nat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5875_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5876_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5877_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5878_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5879_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5880_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5881_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ A2 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5882_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > complex] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups7754918857620584856omplex @ G @ A2 )
          = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups7754918857620584856omplex @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5883_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3542108847815614940at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3542108847815614940at_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5884_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups6591440286371151544t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups6591440286371151544t_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5885_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5886_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5887_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups2073611262835488442omplex @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ A2 ) @ ( groups2073611262835488442omplex @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5888_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5889_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5890_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5891_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5892_ln__add__one__self__le__self,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self
thf(fact_5893_ln__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y2 ) )
          = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).

% ln_mult
thf(fact_5894_ln__eq__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ln_ln_real @ X2 )
          = ( minus_minus_real @ X2 @ one_one_real ) )
       => ( X2 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_5895_ln__div,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y2 ) )
          = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).

% ln_div
thf(fact_5896_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_5897_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_5898_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_5899_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_5900_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_5901_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_5902_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_5903_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_5904_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_5905_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_5906_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_5907_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_5908_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_5909_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_5910_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_5911_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_5912_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_5913_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_5914_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_5915_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_5916_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_5917_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5918_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5919_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5920_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5921_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y2 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5922_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5923_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_5924_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_5925_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_5926_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_5927_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_5928_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_5929_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_5930_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_5931_power2__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_real @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_5932_power2__eq__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_int @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_5933_power2__eq__iff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus1482373934393186551omplex @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_5934_power2__eq__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus1351360451143612070nteger @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_5935_power2__eq__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_rat @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_5936_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_5937_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5938_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5939_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5940_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5941_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5942_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5943_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5944_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5945_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5946_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5947_ln__le__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_5948_ln__diff__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y2 ) @ Y2 ) ) ) ) ).

% ln_diff_le
thf(fact_5949_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_5950_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_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_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_5961_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_5962_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_5963_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_5964_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_5965_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_5966_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_5967_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_5968_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_5969_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_5970_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_5971_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_5972_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_5973_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_5974_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_5975_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_5976_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_5977_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_5978_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_5979_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_5980_realpow__square__minus__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_5981_ln__one__minus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_5982_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_5983_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_5984_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_5985_minus__mod__int__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
        = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5986_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_5987_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5988_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5989_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5990_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5991_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5992_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5993_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5994_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5995_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5996_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B2 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5997_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_5998_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_5999_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_6000_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_6001_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_6002_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_6003_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_6004_square__le__1,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_6005_square__le__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
     => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_6006_square__le__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_6007_square__le__1,axiom,
    ! [X2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
     => ( ( ord_less_eq_int @ X2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_6008_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_6009_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_6010_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_6011_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_6012_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_6013_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_6014_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_6015_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_6016_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_6017_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_6018_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_6019_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_6020_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_6021_div__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_6022_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_6023_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_6024_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_6025_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_6026_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_6027_crossproduct__eq,axiom,
    ! [W: rat,Y2: rat,X2: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y2 ) @ ( times_times_rat @ X2 @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X2 @ Y2 ) ) )
      = ( ( W = X2 )
        | ( Y2 = Z ) ) ) ).

% crossproduct_eq
thf(fact_6028_crossproduct__eq,axiom,
    ! [W: complex,Y2: complex,X2: complex,Z: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y2 ) @ ( times_times_complex @ X2 @ Z ) )
        = ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X2 @ Y2 ) ) )
      = ( ( W = X2 )
        | ( Y2 = Z ) ) ) ).

% crossproduct_eq
thf(fact_6029_crossproduct__eq,axiom,
    ! [W: real,Y2: real,X2: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y2 ) @ ( times_times_real @ X2 @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X2 @ Y2 ) ) )
      = ( ( W = X2 )
        | ( Y2 = Z ) ) ) ).

% crossproduct_eq
thf(fact_6030_crossproduct__eq,axiom,
    ! [W: nat,Y2: nat,X2: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y2 ) @ ( times_times_nat @ X2 @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y2 ) ) )
      = ( ( W = X2 )
        | ( Y2 = Z ) ) ) ).

% crossproduct_eq
thf(fact_6031_crossproduct__eq,axiom,
    ! [W: int,Y2: int,X2: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y2 ) @ ( times_times_int @ X2 @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X2 @ Y2 ) ) )
      = ( ( W = X2 )
        | ( Y2 = Z ) ) ) ).

% crossproduct_eq
thf(fact_6032_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_6033_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_6034_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_6035_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_6036_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_6037_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_6038_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_6039_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_6040_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_6041_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_6042_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_6043_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R4 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_6044_ln__one__plus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_6045_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_6046_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R4 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_6047_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_6048_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_6049_tanh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_6050_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R4: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6051_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R4: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6052_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_6053_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_6054_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( 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 @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_6055_semiring__norm_I90_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(90)
thf(fact_6056_case__prodI2,axiom,
    ! [P6: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P6
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc7828578312038201481er_o_o @ C @ P6 ) ) ).

% case_prodI2
thf(fact_6057_case__prodI2,axiom,
    ! [P6: product_prod_num_num,C: num > num > $o] :
      ( ! [A3: num,B3: num] :
          ( ( P6
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5703948589228662326_num_o @ C @ P6 ) ) ).

% case_prodI2
thf(fact_6058_case__prodI2,axiom,
    ! [P6: product_prod_nat_num,C: nat > num > $o] :
      ( ! [A3: nat,B3: num] :
          ( ( P6
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4927758841916487424_num_o @ C @ P6 ) ) ).

% case_prodI2
thf(fact_6059_case__prodI2,axiom,
    ! [P6: product_prod_int_int,C: int > int > $o] :
      ( ! [A3: int,B3: int] :
          ( ( P6
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4947309494688390418_int_o @ C @ P6 ) ) ).

% case_prodI2
thf(fact_6060_case__prodI2,axiom,
    ! [P6: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P6
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P6 ) ) ).

% case_prodI2
thf(fact_6061_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_6062_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_6063_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_6064_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_6065_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_6066_mem__case__prodI2,axiom,
    ! [P6: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P6
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6067_mem__case__prodI2,axiom,
    ! [P6: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P6
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6068_mem__case__prodI2,axiom,
    ! [P6: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P6
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6069_mem__case__prodI2,axiom,
    ! [P6: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P6
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6070_mem__case__prodI2,axiom,
    ! [P6: product_prod_num_num,Z: nat,C: num > num > set_nat] :
      ( ! [A3: num,B3: num] :
          ( ( P6
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6071_mem__case__prodI2,axiom,
    ! [P6: product_prod_num_num,Z: real,C: num > num > set_real] :
      ( ! [A3: num,B3: num] :
          ( ( P6
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6072_mem__case__prodI2,axiom,
    ! [P6: product_prod_num_num,Z: int,C: num > num > set_int] :
      ( ! [A3: num,B3: num] :
          ( ( P6
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6073_mem__case__prodI2,axiom,
    ! [P6: product_prod_num_num,Z: complex,C: num > num > set_complex] :
      ( ! [A3: num,B3: num] :
          ( ( P6
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6074_mem__case__prodI2,axiom,
    ! [P6: product_prod_nat_num,Z: nat,C: nat > num > set_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P6
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6075_mem__case__prodI2,axiom,
    ! [P6: product_prod_nat_num,Z: real,C: nat > num > set_real] :
      ( ! [A3: nat,B3: num] :
          ( ( P6
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P6 ) ) ) ).

% mem_case_prodI2
thf(fact_6076_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_6077_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_6078_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_6079_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_6080_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_6081_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_6082_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_6083_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_6084_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > num > set_nat,A: nat,B: num] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6085_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_6086_case__prodI2_H,axiom,
    ! [P6: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ( product_Pair_nat_nat @ A3 @ B3 )
            = P6 )
         => ( C @ A3 @ B3 @ X2 ) )
     => ( produc8739625826339149834_nat_o @ C @ P6 @ X2 ) ) ).

% case_prodI2'
thf(fact_6087_semiring__norm_I88_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N2 ) ) ).

% semiring_norm(88)
thf(fact_6088_semiring__norm_I89_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N2 ) ) ).

% semiring_norm(89)
thf(fact_6089_semiring__norm_I84_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit1 @ N2 ) ) ).

% semiring_norm(84)
thf(fact_6090_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_6091_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_numeral
thf(fact_6092_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_numeral
thf(fact_6093_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_numeral
thf(fact_6094_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_numeral
thf(fact_6095_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_6096_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_6097_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_6098_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_6099_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_6100_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_6101_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_6102_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_6103_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_6104_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_6105_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_6106_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_6107_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_6108_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_6109_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_6110_tanh__real__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% tanh_real_less_iff
thf(fact_6111_tanh__real__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% tanh_real_le_iff
thf(fact_6112_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_6113_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_6114_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6115_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6116_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6117_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6118_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_6119_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_6120_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_6121_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_6122_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_6123_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_6124_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_6125_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_6126_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_6127_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_6128_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_6129_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_6130_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_6131_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_6132_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_neg_numeral
thf(fact_6133_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_6134_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_6135_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_6136_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_6137_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_6138_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_6139_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_6140_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_6141_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_6142_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_6143_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_6144_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_6145_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_6146_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_6147_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_6148_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_6149_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_6150_tanh__real__neg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( tanh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_6151_tanh__real__pos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% tanh_real_pos_iff
thf(fact_6152_tanh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_6153_tanh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% tanh_real_nonneg_iff
thf(fact_6154_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_6155_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_6156_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_6157_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_6158_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_6159_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_6160_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_6161_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6162_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_6163_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_6164_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_6165_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_6166_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_6167_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_6168_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_6169_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_6170_artanh__minus__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).

% artanh_minus_real
thf(fact_6171_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_6172_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_6173_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_6174_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_6175_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_6176_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_6177_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_6178_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_6179_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_6180_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_6181_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_6182_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_6183_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_6184_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_6185_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_6186_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_6187_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_6188_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_6189_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_6190_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_6191_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_6192_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_6193_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_6194_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_6195_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_6196_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_6197_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_6198_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_6199_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_6200_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_6201_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_6202_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_6203_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_6204_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_6205_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_6206_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_6207_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_6208_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_6209_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_6210_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_6211_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_6212_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_6213_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_6214_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_6215_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_6216_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_6217_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_6218_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_6219_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_6220_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_6221_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R4: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6222_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R4: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6223_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_6224_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_6225_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_6226_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_6227_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_6228_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_6229_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_6230_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_6231_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_6232_abs__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_mult
thf(fact_6233_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_6234_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_6235_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_6236_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_6237_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_6238_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_6239_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_6240_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_6241_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_6242_mem__case__prodE,axiom,
    ! [Z: nat,C: code_integer > $o > set_nat,P6: produc6271795597528267376eger_o] :
      ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P6 ) )
     => ~ ! [X3: code_integer,Y5: $o] :
            ( ( P6
              = ( produc6677183202524767010eger_o @ X3 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6243_mem__case__prodE,axiom,
    ! [Z: real,C: code_integer > $o > set_real,P6: produc6271795597528267376eger_o] :
      ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P6 ) )
     => ~ ! [X3: code_integer,Y5: $o] :
            ( ( P6
              = ( produc6677183202524767010eger_o @ X3 @ Y5 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6244_mem__case__prodE,axiom,
    ! [Z: int,C: code_integer > $o > set_int,P6: produc6271795597528267376eger_o] :
      ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P6 ) )
     => ~ ! [X3: code_integer,Y5: $o] :
            ( ( P6
              = ( produc6677183202524767010eger_o @ X3 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6245_mem__case__prodE,axiom,
    ! [Z: complex,C: code_integer > $o > set_complex,P6: produc6271795597528267376eger_o] :
      ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P6 ) )
     => ~ ! [X3: code_integer,Y5: $o] :
            ( ( P6
              = ( produc6677183202524767010eger_o @ X3 @ Y5 ) )
           => ~ ( member_complex @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6246_mem__case__prodE,axiom,
    ! [Z: nat,C: num > num > set_nat,P6: product_prod_num_num] :
      ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P6 ) )
     => ~ ! [X3: num,Y5: num] :
            ( ( P6
              = ( product_Pair_num_num @ X3 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6247_mem__case__prodE,axiom,
    ! [Z: real,C: num > num > set_real,P6: product_prod_num_num] :
      ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P6 ) )
     => ~ ! [X3: num,Y5: num] :
            ( ( P6
              = ( product_Pair_num_num @ X3 @ Y5 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6248_mem__case__prodE,axiom,
    ! [Z: int,C: num > num > set_int,P6: product_prod_num_num] :
      ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P6 ) )
     => ~ ! [X3: num,Y5: num] :
            ( ( P6
              = ( product_Pair_num_num @ X3 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6249_mem__case__prodE,axiom,
    ! [Z: complex,C: num > num > set_complex,P6: product_prod_num_num] :
      ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P6 ) )
     => ~ ! [X3: num,Y5: num] :
            ( ( P6
              = ( product_Pair_num_num @ X3 @ Y5 ) )
           => ~ ( member_complex @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6250_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > num > set_nat,P6: product_prod_nat_num] :
      ( ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P6 ) )
     => ~ ! [X3: nat,Y5: num] :
            ( ( P6
              = ( product_Pair_nat_num @ X3 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6251_mem__case__prodE,axiom,
    ! [Z: real,C: nat > num > set_real,P6: product_prod_nat_num] :
      ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P6 ) )
     => ~ ! [X3: nat,Y5: num] :
            ( ( P6
              = ( product_Pair_nat_num @ X3 @ Y5 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6252_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_6253_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_6254_case__prodE,axiom,
    ! [C: code_integer > $o > $o,P6: produc6271795597528267376eger_o] :
      ( ( produc7828578312038201481er_o_o @ C @ P6 )
     => ~ ! [X3: code_integer,Y5: $o] :
            ( ( P6
              = ( produc6677183202524767010eger_o @ X3 @ Y5 ) )
           => ~ ( C @ X3 @ Y5 ) ) ) ).

% case_prodE
thf(fact_6255_case__prodE,axiom,
    ! [C: num > num > $o,P6: product_prod_num_num] :
      ( ( produc5703948589228662326_num_o @ C @ P6 )
     => ~ ! [X3: num,Y5: num] :
            ( ( P6
              = ( product_Pair_num_num @ X3 @ Y5 ) )
           => ~ ( C @ X3 @ Y5 ) ) ) ).

% case_prodE
thf(fact_6256_case__prodE,axiom,
    ! [C: nat > num > $o,P6: product_prod_nat_num] :
      ( ( produc4927758841916487424_num_o @ C @ P6 )
     => ~ ! [X3: nat,Y5: num] :
            ( ( P6
              = ( product_Pair_nat_num @ X3 @ Y5 ) )
           => ~ ( C @ X3 @ Y5 ) ) ) ).

% case_prodE
thf(fact_6257_case__prodE,axiom,
    ! [C: int > int > $o,P6: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P6 )
     => ~ ! [X3: int,Y5: int] :
            ( ( P6
              = ( product_Pair_int_int @ X3 @ Y5 ) )
           => ~ ( C @ X3 @ Y5 ) ) ) ).

% case_prodE
thf(fact_6258_case__prodE,axiom,
    ! [C: nat > nat > $o,P6: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P6 )
     => ~ ! [X3: nat,Y5: nat] :
            ( ( P6
              = ( product_Pair_nat_nat @ X3 @ Y5 ) )
           => ~ ( C @ X3 @ Y5 ) ) ) ).

% case_prodE
thf(fact_6259_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_6260_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_6261_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_6262_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_6263_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_6264_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P6: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P6 @ Z )
     => ~ ! [X3: nat,Y5: nat] :
            ( ( P6
              = ( product_Pair_nat_nat @ X3 @ Y5 ) )
           => ~ ( C @ X3 @ Y5 @ Z ) ) ) ).

% case_prodE'
thf(fact_6265_case__prodD_H,axiom,
    ! [R2: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R2 @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R2 @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_6266_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_6267_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_6268_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_6269_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_6270_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6271_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6272_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6273_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6274_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_6275_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_6276_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_6277_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_6278_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_6279_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_6280_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_6281_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_6282_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_6283_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_6284_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_6285_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_6286_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_6287_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_6288_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_6289_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_6290_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_6291_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_6292_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_6293_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_6294_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_6295_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_6296_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_6297_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_6298_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_6299_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_6300_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_6301_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_6302_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_6303_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_6304_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_6305_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_6306_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_6307_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_6308_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_6309_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_6310_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_6311_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_6312_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_6313_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_6314_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_6315_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_6316_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_6317_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_6318_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_6319_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_6320_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_6321_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_6322_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_real_def
thf(fact_6323_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one )
     => ( ! [X23: num] :
            ( Y2
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y2
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_6324_xor__num_Ocases,axiom,
    ! [X2: product_prod_num_num] :
      ( ( X2
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N4: num] :
            ( X2
           != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
       => ( ! [N4: num] :
              ( X2
             != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
         => ( ! [M5: num] :
                ( X2
               != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
           => ( ! [M5: num,N4: num] :
                  ( X2
                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N4 ) ) )
             => ( ! [M5: num,N4: num] :
                    ( X2
                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N4 ) ) )
               => ( ! [M5: num] :
                      ( X2
                     != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
                 => ( ! [M5: num,N4: num] :
                        ( X2
                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N4 ) ) )
                   => ~ ! [M5: num,N4: num] :
                          ( X2
                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_6325_sin__bound__lemma,axiom,
    ! [X2: real,Y2: real,U: real,V: real] :
      ( ( X2 = Y2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y2 ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_6326_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
          @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6327_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
          @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6328_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
          @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6329_sum__subtractf__nat,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups977919841031483927at_nat
          @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6330_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
          @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6331_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_6332_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_6333_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_6334_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_6335_tanh__real__lt__1,axiom,
    ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_6336_tanh__real__gt__neg1,axiom,
    ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).

% tanh_real_gt_neg1
thf(fact_6337_dense__eq0__I,axiom,
    ! [X2: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E2 ) )
     => ( X2 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_6338_dense__eq0__I,axiom,
    ! [X2: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E2 ) )
     => ( X2 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_6339_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_6340_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_6341_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_6342_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_6343_abs__mult__pos,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y2 ) @ X2 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_6344_abs__mult__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( times_times_real @ ( abs_abs_real @ Y2 ) @ X2 )
        = ( abs_abs_real @ ( times_times_real @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_6345_abs__mult__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y2 ) @ X2 )
        = ( abs_abs_rat @ ( times_times_rat @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_6346_abs__mult__pos,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( times_times_int @ ( abs_abs_int @ Y2 ) @ X2 )
        = ( abs_abs_int @ ( times_times_int @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_6347_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_6348_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6349_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_6350_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_6351_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_6352_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_6353_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_6354_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_6355_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_6356_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_6357_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_6358_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_6359_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_6360_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_6361_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_6362_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_6363_abs__div__pos,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y2 )
        = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% abs_div_pos
thf(fact_6364_abs__div__pos,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y2 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% abs_div_pos
thf(fact_6365_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6366_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6367_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6368_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6369_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_6370_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_6371_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_6372_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_6373_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6374_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6375_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6376_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6377_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_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_abs__diff__le__iff,axiom,
    ! [X2: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A ) ) @ R )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X2 )
        & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6386_abs__diff__le__iff,axiom,
    ! [X2: real,A: real,R: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X2 )
        & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6387_abs__diff__le__iff,axiom,
    ! [X2: rat,A: rat,R: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A ) ) @ R )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X2 )
        & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6388_abs__diff__le__iff,axiom,
    ! [X2: int,A: int,R: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X2 )
        & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6389_abs__diff__less__iff,axiom,
    ! [X2: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A ) ) @ R )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X2 )
        & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6390_abs__diff__less__iff,axiom,
    ! [X2: real,A: real,R: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A ) ) @ R )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X2 )
        & ( ord_less_real @ X2 @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6391_abs__diff__less__iff,axiom,
    ! [X2: rat,A: rat,R: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A ) ) @ R )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X2 )
        & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6392_abs__diff__less__iff,axiom,
    ! [X2: int,A: int,R: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A ) ) @ R )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X2 )
        & ( ord_less_int @ X2 @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6393_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6394_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6395_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_6396_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6397_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6398_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_6399_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_6400_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_6401_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_6402_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_6403_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_6404_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6405_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6406_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6407_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6408_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6409_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6410_power__minus__Bit1,axiom,
    ! [X2: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6411_power__minus__Bit1,axiom,
    ! [X2: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6412_power__minus__Bit1,axiom,
    ! [X2: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6413_power__minus__Bit1,axiom,
    ! [X2: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6414_power__minus__Bit1,axiom,
    ! [X2: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6415_lemma__interval__lt,axiom,
    ! [A: real,X2: real,B: real] :
      ( ( ord_less_real @ A @ X2 )
     => ( ( ord_less_real @ X2 @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y3: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y3 )
                  & ( ord_less_real @ Y3 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_6416_sum__power__add,axiom,
    ! [X2: complex,M: nat,I6: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I3 ) )
        @ I6 )
      = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_6417_sum__power__add,axiom,
    ! [X2: int,M: nat,I6: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I3 ) )
        @ I6 )
      = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_6418_sum__power__add,axiom,
    ! [X2: real,M: nat,I6: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I3 ) )
        @ I6 )
      = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_6419_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_6420_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_6421_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_6422_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_6423_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_6424_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_6425_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_6426_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_6427_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_6428_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_6429_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_6430_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_6431_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_6432_abs__add__one__gt__zero,axiom,
    ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6433_abs__add__one__gt__zero,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6434_abs__add__one__gt__zero,axiom,
    ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6435_abs__add__one__gt__zero,axiom,
    ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6436_sum__diff__nat,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6437_sum__diff__nat,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6438_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_6439_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_6440_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_6441_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_6442_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_6443_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_6444_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_6445_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_6446_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_6447_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_6448_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_6449_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_6450_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_6451_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_6452_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_6453_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_6454_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_6455_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_6456_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_6457_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6458_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6459_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_6460_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6461_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6462_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_6463_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_6464_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_6465_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_6466_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_6467_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_6468_lemma__interval,axiom,
    ! [A: real,X2: real,B: real] :
      ( ( ord_less_real @ A @ X2 )
     => ( ( ord_less_real @ X2 @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y3: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y3 )
                  & ( ord_less_eq_real @ Y3 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_6469_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_6470_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_6471_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_6472_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_6473_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > complex] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( minus_minus_complex @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_complex @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6474_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_6475_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_6476_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_6477_abs__le__square__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y2 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6478_abs__le__square__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y2 ) )
      = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6479_abs__le__square__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y2 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6480_abs__le__square__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6481_abs__square__eq__1,axiom,
    ! [X2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X2 )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_6482_abs__square__eq__1,axiom,
    ! [X2: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X2 )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_6483_abs__square__eq__1,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X2 )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_6484_abs__square__eq__1,axiom,
    ! [X2: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X2 )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_6485_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_6486_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_6487_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_6488_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_6489_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( 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 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6490_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( 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 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6491_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( 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 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6492_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( 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 @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6493_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6494_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6495_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6496_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6497_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_6498_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_6499_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_6500_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X2: 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 @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6501_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X2: 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 @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6502_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X2: 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 @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6503_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X2: 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 @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6504_power2__le__iff__abs__le,axiom,
    ! [Y2: code_integer,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6505_power2__le__iff__abs__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6506_power2__le__iff__abs__le,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6507_power2__le__iff__abs__le,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6508_abs__square__le__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_6509_abs__square__le__1,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_6510_abs__square__le__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_6511_abs__square__le__1,axiom,
    ! [X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_6512_abs__square__less__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_6513_abs__square__less__1,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_6514_abs__square__less__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_6515_abs__square__less__1,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_6516_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_6517_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_6518_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_6519_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_6520_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_6521_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_6522_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_6523_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X2: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I4 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X2 @ I6 )
          = one_one_Code_integer )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6524_convex__sum__bound__le,axiom,
    ! [I6: set_real,X2: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I4 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X2 @ I6 )
          = one_one_Code_integer )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6525_convex__sum__bound__le,axiom,
    ! [I6: set_int,X2: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I4 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X2 @ I6 )
          = one_one_Code_integer )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6526_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X2: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I4 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X2 @ I6 )
          = one_one_Code_integer )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6527_convex__sum__bound__le,axiom,
    ! [I6: set_real,X2: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I4 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X2 @ I6 )
          = one_one_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6528_convex__sum__bound__le,axiom,
    ! [I6: set_int,X2: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I4 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X2 @ I6 )
          = one_one_real )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6529_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X2: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I4 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X2 @ I6 )
          = one_one_real )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6530_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X2: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I4 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X2 @ I6 )
          = one_one_rat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6531_convex__sum__bound__le,axiom,
    ! [I6: set_real,X2: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I4 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X2 @ I6 )
          = one_one_rat )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6532_convex__sum__bound__le,axiom,
    ! [I6: set_int,X2: int > rat,A: int > rat,B: rat,Delta: rat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I4 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X2 @ I6 )
          = one_one_rat )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X2 @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6533_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ 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
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_6534_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ 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
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_6535_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ 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
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_6536_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ 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
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_6537_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2073611262835488442omplex
          @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_complex @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6538_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6539_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6540_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N ) @ ( modulo_modulo_nat @ M6 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_6541_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_6542_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_6543_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6544_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6545_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6546_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6547_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_6548_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_6549_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_6550_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_6551_eq__diff__eq_H,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( X2
        = ( minus_minus_real @ Y2 @ Z ) )
      = ( Y2
        = ( plus_plus_real @ X2 @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_6552_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_6553_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_6554_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6555_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_6556_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_6557_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_6558_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6559_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6560_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6561_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_6562_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_6563_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( 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_6564_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_6565_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_6566_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6567_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6568_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_6569_signed__take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_6570_arctan__double,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_6571_zdvd1__eq,axiom,
    ! [X2: int] :
      ( ( dvd_dvd_int @ X2 @ one_one_int )
      = ( ( abs_abs_int @ X2 )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_6572_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_6573_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_6574_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_6575_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_6576_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_6577_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6578_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_6579_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_6580_zero__less__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% zero_less_arctan_iff
thf(fact_6581_arctan__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( arctan @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_6582_arctan__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_6583_zero__le__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_arctan_iff
thf(fact_6584_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_6585_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_6586_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_6587_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_6588_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_6589_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_6590_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_6591_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_6592_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_6593_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_6594_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_6595_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_6596_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_6597_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_6598_signed__take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_6599_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_6600_arctan__monotone,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) ) ) ).

% arctan_monotone
thf(fact_6601_arctan__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% arctan_less_iff
thf(fact_6602_arctan__monotone_H,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) ) ) ).

% arctan_monotone'
thf(fact_6603_arctan__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% arctan_le_iff
thf(fact_6604_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_6605_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6606_abs__mod__less,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).

% abs_mod_less
thf(fact_6607_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_6608_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_6609_even__add__abs__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_add_abs_iff
thf(fact_6610_even__abs__add__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_abs_add_iff
thf(fact_6611_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ( ord_less_eq_nat @ M @ I4 )
            & ( ord_less_nat @ I4 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I4: nat] :
                ( ( ord_less_eq_nat @ M @ I4 )
                & ( ord_less_eq_nat @ I4 @ N2 )
                & ( ( F @ I4 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_6612_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_6613_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_6614_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_6615_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_6616_incr__lemma,axiom,
    ! [D: int,Z: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_6617_decr__lemma,axiom,
    ! [D: int,X2: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_6618_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_6619_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_6620_arctan__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y2 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_6621_sum__gp,axiom,
    ! [N2: nat,M: nat,X2: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X2 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6622_sum__gp,axiom,
    ! [N2: nat,M: nat,X2: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X2 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6623_sum__gp,axiom,
    ! [N2: nat,M: nat,X2: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X2 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6624_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6625_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6626_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6627_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6628_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_6629_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_6630_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_6631_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6632_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6633_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6634_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6635_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6636_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_6637_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_6638_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6639_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6640_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_6641_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_6642_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_6643_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_6644_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_6645_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_6646_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_6647_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_6648_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_6649_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_6650_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_6651_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6652_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_6653_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_6654_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_6655_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_6656_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_6657_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_6658_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_6659_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_6660_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_6661_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_6662_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_6663_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_6664_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_6665_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_6666_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6690914467698888265omplex @ N2 ) ) ).

% of_nat_numeral
thf(fact_6667_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% of_nat_numeral
thf(fact_6668_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% of_nat_numeral
thf(fact_6669_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% of_nat_numeral
thf(fact_6670_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% of_nat_numeral
thf(fact_6671_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_6672_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_6673_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_6674_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_6675_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_6676_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_6677_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_6678_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_6679_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_6680_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_6681_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_6682_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_6683_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_6684_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_6685_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_6686_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_6687_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_6688_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_6689_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_6690_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_6691_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_6692_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_6693_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_6694_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6695_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_6696_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_6697_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_6698_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_6699_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6700_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_6701_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_6702_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_6703_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_6704_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_6705_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_6706_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_6707_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_6708_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_6709_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_6710_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_6711_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_6712_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_6713_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_6714_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_6715_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X2 )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6716_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X2 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6717_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri681578069525770553at_rat @ X2 )
        = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6718_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X2 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6719_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X2 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X2
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6720_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6721_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6722_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
        = ( semiri681578069525770553at_rat @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6723_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6724_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X2 ) )
      = ( ( power_power_nat @ B @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6725_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_6726_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_6727_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_6728_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_6729_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_6730_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_6731_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_6732_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_6733_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_6734_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_6735_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_6736_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_6737_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_6738_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_6739_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_6740_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_6741_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_6742_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_mult
thf(fact_6743_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_6744_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_6745_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_6746_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_6747_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_6748_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_6749_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_6750_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_6751_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_6752_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_6753_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6754_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6755_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6756_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X2 ) )
      = ( ( power_power_int @ B @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6757_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X2 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6758_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X2 )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6759_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X2 )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6760_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X2 )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X2
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6761_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_6762_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_6763_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_6764_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_6765_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_6766_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_6767_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_nat_of_bool
thf(fact_6768_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_6769_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_6770_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_6771_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_6772_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_6773_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_6774_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_6775_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_6776_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6777_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_6778_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_6779_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_6780_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_6781_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_6782_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_6783_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_6784_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_6785_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_6786_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_6787_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_6788_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_6789_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_6790_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_6791_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_6792_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_6793_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_6794_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_6795_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_6796_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_6797_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_6798_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_6799_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_6800_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_6801_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_6802_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_6803_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6804_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6805_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_6806_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y2 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6807_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y2 )
        = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6808_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y2 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6809_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y2 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6810_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y2 )
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6811_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 )
        = ( semiri8010041392384452111omplex @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6812_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 )
        = ( semiri5074537144036343181t_real @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6813_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 )
        = ( semiri681578069525770553at_rat @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6814_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = ( semiri1316708129612266289at_nat @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6815_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = ( semiri1314217659103216013at_int @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6816_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6817_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6818_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6819_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6820_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6821_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6822_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6823_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6824_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6825_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6826_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6827_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6828_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6829_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6830_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6831_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6832_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_6833_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_6834_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_6835_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_6836_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_6837_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_6838_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_6839_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_6840_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_6841_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_6842_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_6843_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_6844_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_6845_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_6846_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_6847_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_6848_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_6849_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_6850_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_6851_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_6852_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_6853_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_6854_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_6855_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_6856_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_6857_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_6858_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_6859_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_6860_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_6861_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_6862_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_6863_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_6864_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_6865_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_6866_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_6867_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_6868_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y2 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6869_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y2 )
        = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6870_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y2 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6871_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y2 )
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6872_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6873_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 )
        = ( ring_1_of_int_real @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6874_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 )
        = ( ring_1_of_int_rat @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6875_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = ( ring_1_of_int_int @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6876_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6877_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6878_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6879_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6880_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6881_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6882_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6883_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6884_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6885_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6886_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6887_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6888_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6889_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6890_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6891_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6892_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6893_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6894_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6895_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6896_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6897_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6898_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6899_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6900_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6901_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6902_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6903_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6904_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6905_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6906_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6907_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6908_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_6909_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_6910_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_6911_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6912_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6913_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6914_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6915_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6916_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6917_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6918_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6919_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6920_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6921_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6922_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6923_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 )
        = ( ring_1_of_int_real @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6924_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = ( ring_1_of_int_int @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6925_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6926_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6927_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 )
        = ( ring_1_of_int_rat @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6928_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y2 )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6929_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y2 )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6930_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y2 )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6931_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y2 )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6932_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y2 )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6933_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6934_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6935_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6936_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6937_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6938_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6939_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6940_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6941_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6942_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6943_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6944_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6945_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6946_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6947_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6948_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6949_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_6950_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_6951_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_6952_mult__of__int__commute,axiom,
    ! [X2: int,Y2: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X2 ) @ Y2 )
      = ( times_times_rat @ Y2 @ ( ring_1_of_int_rat @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_6953_mult__of__int__commute,axiom,
    ! [X2: int,Y2: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X2 ) @ Y2 )
      = ( times_times_complex @ Y2 @ ( ring_17405671764205052669omplex @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_6954_mult__of__int__commute,axiom,
    ! [X2: int,Y2: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ Y2 )
      = ( times_times_real @ Y2 @ ( ring_1_of_int_real @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_6955_mult__of__int__commute,axiom,
    ! [X2: int,Y2: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X2 ) @ Y2 )
      = ( times_times_int @ Y2 @ ( ring_1_of_int_int @ X2 ) ) ) ).

% mult_of_int_commute
thf(fact_6956_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X2 ) @ Y2 )
      = ( times_times_complex @ Y2 @ ( semiri8010041392384452111omplex @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_6957_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y2 )
      = ( times_times_real @ Y2 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_6958_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y2 )
      = ( times_times_rat @ Y2 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_6959_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y2 )
      = ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_6960_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y2 )
      = ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_6961_of__int__max,axiom,
    ! [X2: int,Y2: int] :
      ( ( ring_1_of_int_real @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ord_max_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ Y2 ) ) ) ).

% of_int_max
thf(fact_6962_of__int__max,axiom,
    ! [X2: int,Y2: int] :
      ( ( ring_1_of_int_rat @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ord_max_rat @ ( ring_1_of_int_rat @ X2 ) @ ( ring_1_of_int_rat @ Y2 ) ) ) ).

% of_int_max
thf(fact_6963_of__int__max,axiom,
    ! [X2: int,Y2: int] :
      ( ( ring_1_of_int_int @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ord_max_int @ ( ring_1_of_int_int @ X2 ) @ ( ring_1_of_int_int @ Y2 ) ) ) ).

% of_int_max
thf(fact_6964_of__int__max,axiom,
    ! [X2: int,Y2: int] :
      ( ( ring_18347121197199848620nteger @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ord_max_Code_integer @ ( ring_18347121197199848620nteger @ X2 ) @ ( ring_18347121197199848620nteger @ Y2 ) ) ) ).

% of_int_max
thf(fact_6965_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_6966_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_6967_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_6968_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_6969_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_6970_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_6971_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_6972_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_6973_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_6974_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_6975_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_6976_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_6977_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_6978_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_6979_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_6980_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_6981_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_6982_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_6983_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_6984_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_6985_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_6986_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_6987_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_6988_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_6989_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_6990_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_6991_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_6992_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_6993_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_6994_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_6995_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_6996_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_6997_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_6998_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_6999_int__cases,axiom,
    ! [Z: int] :
      ( ! [N4: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% int_cases
thf(fact_7000_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
     => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_7001_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_7002_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_7003_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_7004_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_7005_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_7006_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_7007_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_7008_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_7009_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_7010_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_7011_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_7012_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X2 ) @ ( semiri4216267220026989637d_enat @ Y2 ) ) ) ).

% of_nat_max
thf(fact_7013_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X2 ) @ ( semiri4939895301339042750nteger @ Y2 ) ) ) ).

% of_nat_max
thf(fact_7014_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y2 ) ) ) ).

% of_nat_max
thf(fact_7015_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( semiri681578069525770553at_rat @ Y2 ) ) ) ).

% of_nat_max
thf(fact_7016_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y2 ) ) ) ).

% of_nat_max
thf(fact_7017_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) ).

% of_nat_max
thf(fact_7018_semiring__norm_I27_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit0 @ N2 ) )
      = ( bit1 @ ( bitM @ N2 ) ) ) ).

% semiring_norm(27)
thf(fact_7019_semiring__norm_I28_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ N2 ) ) ) ).

% semiring_norm(28)
thf(fact_7020_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_7021_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_7022_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_7023_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_7024_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_7025_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_7026_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_7027_reals__Archimedean3,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ! [Y3: real] :
        ? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X2 ) ) ) ).

% reals_Archimedean3
thf(fact_7028_real__of__int__div4,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) ) ) ).

% real_of_int_div4
thf(fact_7029_int__cases4,axiom,
    ! [M: int] :
      ( ! [N4: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% int_cases4
thf(fact_7030_real__of__nat__div4,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% real_of_nat_div4
thf(fact_7031_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_7032_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_7033_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z5: int] :
        ? [N: nat] :
          ( Z5
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_7034_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_7035_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_7036_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_7037_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_7038_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_7039_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_7040_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_7041_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_7042_of__int__leD,axiom,
    ! [N2: int,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).

% of_int_leD
thf(fact_7043_of__int__leD,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% of_int_leD
thf(fact_7044_of__int__leD,axiom,
    ! [N2: int,X2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).

% of_int_leD
thf(fact_7045_of__int__leD,axiom,
    ! [N2: int,X2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).

% of_int_leD
thf(fact_7046_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_7047_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_7048_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_7049_of__int__lessD,axiom,
    ! [N2: int,X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7050_of__int__lessD,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7051_of__int__lessD,axiom,
    ! [N2: int,X2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7052_of__int__lessD,axiom,
    ! [N2: int,X2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7053_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_7054_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_7055_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_7056_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_7057_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_7058_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_7059_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_7060_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_7061_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_7062_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_7063_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_7064_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).

% int_less_real_le
thf(fact_7065_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N4 )
          & ( K
            = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_7066_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N4: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).

% pos_int_cases
thf(fact_7067_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N4: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
       => ~ ! [N4: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% int_cases3
thf(fact_7068_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).

% nat_less_real_le
thf(fact_7069_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_7070_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_7071_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_7072_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7073_negD,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ zero_zero_int )
     => ? [N4: nat] :
          ( X2
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% negD
thf(fact_7074_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_7075_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_7076_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_7077_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_7078_real__of__int__div__aux,axiom,
    ! [X2: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X2 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X2 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_7079_real__of__nat__div__aux,axiom,
    ! [X2: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X2 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X2 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_7080_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_7081_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_7082_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_7083_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_7084_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7085_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7086_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7087_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_7088_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_7089_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_7090_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_7091_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_7092_real__archimedian__rdiv__eq__0,axiom,
    ! [X2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M5: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M5 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X2 ) @ C ) )
         => ( X2 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_7093_real__of__int__div2,axiom,
    ! [N2: int,X2: 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 @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X2 ) ) ) ) ).

% real_of_int_div2
thf(fact_7094_real__of__int__div3,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X2 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_7095_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N4: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).

% neg_int_cases
thf(fact_7096_zdiff__int__split,axiom,
    ! [P: int > $o,X2: nat,Y2: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y2 ) ) )
      = ( ( ( ord_less_eq_nat @ Y2 @ X2 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
        & ( ( ord_less_nat @ X2 @ Y2 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_7097_real__of__nat__div2,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) ) ) ).

% real_of_nat_div2
thf(fact_7098_real__of__nat__div3,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_7099_ln__realpow,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( power_power_real @ X2 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_realpow
thf(fact_7100_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_7101_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_7102_linear__plus__1__le__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N2 ) ) ) ).

% linear_plus_1_le_power
thf(fact_7103_Bernoulli__inequality,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N2 ) ) ) ).

% Bernoulli_inequality
thf(fact_7104_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X2: 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 ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N2 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_7105_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_7106_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_7107_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_7108_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_7109_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_7110_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_7111_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_7112_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_7113_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_7114_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_7115_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_7116_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_7117_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_7118_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_7119_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_7120_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_7121_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_7122_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_7123_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_7124_sum__gp__offset,axiom,
    ! [X2: complex,M: nat,N2: nat] :
      ( ( ( X2 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X2 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7125_sum__gp__offset,axiom,
    ! [X2: rat,M: nat,N2: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7126_sum__gp__offset,axiom,
    ! [X2: real,M: nat,N2: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7127_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M6: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7128_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M6: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7129_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M6: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7130_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M6: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7131_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M6: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7132_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7133_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N4: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7134_floor__exists1,axiom,
    ! [X2: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y3: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ X2 )
            & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
thf(fact_7135_floor__exists1,axiom,
    ! [X2: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y3: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ X2 )
            & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y3 @ one_one_int ) ) ) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
thf(fact_7136_floor__exists,axiom,
    ! [X2: real] :
    ? [Z4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7137_floor__exists,axiom,
    ! [X2: rat] :
    ? [Z4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7138_monoseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7139_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_7140_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_7141_complex__mod__minus__le__complex__mod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_7142_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_7143_monoseq__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).

% monoseq_realpow
thf(fact_7144_real__arch__simple,axiom,
    ! [X2: real] :
    ? [N4: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% real_arch_simple
thf(fact_7145_real__arch__simple,axiom,
    ! [X2: rat] :
    ? [N4: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% real_arch_simple
thf(fact_7146_reals__Archimedean2,axiom,
    ! [X2: real] :
    ? [N4: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% reals_Archimedean2
thf(fact_7147_reals__Archimedean2,axiom,
    ! [X2: rat] :
    ? [N4: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% reals_Archimedean2
thf(fact_7148_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N4: nat] :
            ( ~ ( P @ N4 )
            & ( P @ ( suc @ N4 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_7149_ex__le__of__int,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_le_of_int
thf(fact_7150_ex__le__of__int,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_le_of_int
thf(fact_7151_ex__less__of__int,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_less_of_int
thf(fact_7152_ex__less__of__int,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_less_of_int
thf(fact_7153_ex__of__int__less,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X2 ) ).

% ex_of_int_less
thf(fact_7154_ex__of__int__less,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 ) ).

% ex_of_int_less
thf(fact_7155_ex__less__of__nat__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [N4: nat] : ( ord_less_real @ Y2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X2 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7156_ex__less__of__nat__mult,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ? [N4: nat] : ( ord_less_rat @ Y2 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X2 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7157_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_7158_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_7159_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_7160_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_7161_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_7162_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_7163_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_7164_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7165_norm__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_7166_norm__le__zero__iff,axiom,
    ! [X2: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
      = ( X2 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_7167_zero__less__norm__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
      = ( X2 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_7168_zero__less__norm__iff,axiom,
    ! [X2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
      = ( X2 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_7169_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_7170_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_7171_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7172_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7173_norm__not__less__zero,axiom,
    ! [X2: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_7174_norm__ge__zero,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% norm_ge_zero
thf(fact_7175_norm__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).

% norm_mult
thf(fact_7176_norm__mult,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).

% norm_mult
thf(fact_7177_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_7178_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_7179_sum__norm__le,axiom,
    ! [S: set_real,F: real > complex,G: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).

% sum_norm_le
thf(fact_7180_sum__norm__le,axiom,
    ! [S: set_int,F: int > complex,G: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ).

% sum_norm_le
thf(fact_7181_sum__norm__le,axiom,
    ! [S: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > complex,G: product_prod_nat_nat > real] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6381953495645901045omplex @ F @ S ) ) @ ( groups4567486121110086003t_real @ G @ S ) ) ) ).

% sum_norm_le
thf(fact_7182_sum__norm__le,axiom,
    ! [S: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).

% sum_norm_le
thf(fact_7183_sum__norm__le,axiom,
    ! [S: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ).

% sum_norm_le
thf(fact_7184_sum__norm__le,axiom,
    ! [S: set_nat,F: nat > real,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).

% sum_norm_le
thf(fact_7185_norm__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N2 ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N2 ) ) ).

% norm_power
thf(fact_7186_norm__power,axiom,
    ! [X2: complex,N2: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N2 ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N2 ) ) ).

% norm_power
thf(fact_7187_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_7188_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_7189_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_7190_norm__uminus__minus,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) ) ).

% norm_uminus_minus
thf(fact_7191_norm__uminus__minus,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ Y2 ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) ) ).

% norm_uminus_minus
thf(fact_7192_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_7193_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_7194_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_7195_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_7196_norm__mult__less,axiom,
    ! [X2: real,R: real,Y2: real,S3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) ) @ ( times_times_real @ R @ S3 ) ) ) ) ).

% norm_mult_less
thf(fact_7197_norm__mult__less,axiom,
    ! [X2: complex,R: real,Y2: complex,S3: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) ) @ ( times_times_real @ R @ S3 ) ) ) ) ).

% norm_mult_less
thf(fact_7198_norm__mult__ineq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).

% norm_mult_ineq
thf(fact_7199_norm__mult__ineq,axiom,
    ! [X2: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).

% norm_mult_ineq
thf(fact_7200_norm__add__less,axiom,
    ! [X2: real,R: real,Y2: real,S3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).

% norm_add_less
thf(fact_7201_norm__add__less,axiom,
    ! [X2: complex,R: real,Y2: complex,S3: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S3 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).

% norm_add_less
thf(fact_7202_norm__triangle__lt,axiom,
    ! [X2: real,Y2: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7203_norm__triangle__lt,axiom,
    ! [X2: complex,Y2: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7204_norm__power__ineq,axiom,
    ! [X2: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_7205_norm__power__ineq,axiom,
    ! [X2: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_7206_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_7207_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_7208_norm__triangle__le,axiom,
    ! [X2: real,Y2: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7209_norm__triangle__le,axiom,
    ! [X2: complex,Y2: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7210_norm__triangle__ineq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) ) ).

% norm_triangle_ineq
thf(fact_7211_norm__triangle__ineq,axiom,
    ! [X2: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) ) ).

% norm_triangle_ineq
thf(fact_7212_norm__triangle__mono,axiom,
    ! [A: real,R: real,B: real,S3: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S3 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).

% norm_triangle_mono
thf(fact_7213_norm__triangle__mono,axiom,
    ! [A: complex,R: real,B: complex,S3: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S3 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).

% norm_triangle_mono
thf(fact_7214_norm__diff__triangle__less,axiom,
    ! [X2: real,Y2: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7215_norm__diff__triangle__less,axiom,
    ! [X2: complex,Y2: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7216_norm__triangle__sub,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) ) ) ).

% norm_triangle_sub
thf(fact_7217_norm__triangle__sub,axiom,
    ! [X2: complex,Y2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) ) ) ).

% norm_triangle_sub
thf(fact_7218_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_7219_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_7220_norm__diff__triangle__le,axiom,
    ! [X2: real,Y2: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7221_norm__diff__triangle__le,axiom,
    ! [X2: complex,Y2: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7222_norm__triangle__le__diff,axiom,
    ! [X2: real,Y2: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_7223_norm__triangle__le__diff,axiom,
    ! [X2: complex,Y2: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_7224_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_7225_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_7226_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_7227_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_7228_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_7229_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_7230_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_7231_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_7232_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_7233_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_7234_square__norm__one,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X2 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_7235_square__norm__one,axiom,
    ! [X2: complex] :
      ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X2 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_7236_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_7237_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_7238_ln__series,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X2 )
          = ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7239_arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arctan @ X2 )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7240_round__unique,axiom,
    ! [X2: real,Y2: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y2 ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y2 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X2 )
          = Y2 ) ) ) ).

% round_unique
thf(fact_7241_round__unique,axiom,
    ! [X2: rat,Y2: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y2 ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y2 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X2 )
          = Y2 ) ) ) ).

% round_unique
thf(fact_7242_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
            @ ^ [P4: 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 ) ) @ P4 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7243_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
            @ ^ [P4: 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 ) ) @ P4 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7244_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
            @ ^ [P4: 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 ) ) @ P4 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7245_summable__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7246_round__unique_H,axiom,
    ! [X2: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X2 )
        = N2 ) ) ).

% round_unique'
thf(fact_7247_round__unique_H,axiom,
    ! [X2: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X2 )
        = N2 ) ) ).

% round_unique'
thf(fact_7248_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7249_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_7250_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_7251_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7252_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_7253_lessThan__subset__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y2 ) )
      = ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_7254_lessThan__subset__iff,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y2 ) )
      = ( ord_less_eq_num @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_7255_lessThan__subset__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y2 ) )
      = ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_7256_lessThan__subset__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y2 ) )
      = ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_7257_lessThan__subset__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% lessThan_subset_iff
thf(fact_7258_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_7259_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_7260_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_7261_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_7262_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_7263_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_7264_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_7265_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_7266_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_7267_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_7268_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7269_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7270_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7271_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7272_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7273_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7274_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7275_powser__insidea,axiom,
    ! [F: nat > real,X2: real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7276_powser__insidea,axiom,
    ! [F: nat > complex,X2: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7277_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_7278_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_7279_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_7280_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_7281_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_7282_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_7283_round__mono,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y2 ) ) ) ).

% round_mono
thf(fact_7284_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_7285_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_7286_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
          @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7287_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
          @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7288_powser__inside,axiom,
    ! [F: nat > real,X2: real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ).

% powser_inside
thf(fact_7289_powser__inside,axiom,
    ! [F: nat > complex,X2: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ).

% powser_inside
thf(fact_7290_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_7291_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_7292_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_7293_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_7294_sum__lessThan__telescope_H,axiom,
    ! [F: nat > complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7295_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7296_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7297_sum__lessThan__telescope,axiom,
    ! [F: nat > complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [N: nat] : ( minus_minus_complex @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_complex @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7298_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7299_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7300_sumr__diff__mult__const2,axiom,
    ! [F: nat > complex,N2: nat,R: complex] :
      ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ R ) )
      = ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( minus_minus_complex @ ( F @ I3 ) @ R )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7301_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N2: nat,R: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ R ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7302_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N2: nat,R: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ R ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7303_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N2: nat,R: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ R ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7304_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_7305_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_7306_power__diff__1__eq,axiom,
    ! [X2: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7307_power__diff__1__eq,axiom,
    ! [X2: complex,N2: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N2 ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X2 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7308_power__diff__1__eq,axiom,
    ! [X2: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7309_power__diff__1__eq,axiom,
    ! [X2: real,N2: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ N2 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_7310_one__diff__power__eq,axiom,
    ! [X2: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7311_one__diff__power__eq,axiom,
    ! [X2: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7312_one__diff__power__eq,axiom,
    ! [X2: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7313_one__diff__power__eq,axiom,
    ! [X2: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_7314_geometric__sum,axiom,
    ! [X2: complex,N2: nat] :
      ( ( X2 != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_7315_geometric__sum,axiom,
    ! [X2: rat,N2: nat] :
      ( ( X2 != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_7316_geometric__sum,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2 != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_7317_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_7318_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_7319_sum__gp__strict,axiom,
    ! [X2: complex,N2: nat] :
      ( ( ( X2 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ N2 ) ) )
      & ( ( X2 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7320_sum__gp__strict,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7321_sum__gp__strict,axiom,
    ! [X2: real,N2: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ N2 ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7322_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ P4 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7323_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ P4 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7324_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ P4 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7325_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ P4 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7326_diff__power__eq__sum,axiom,
    ! [X2: complex,N2: nat,Y2: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) @ ( power_power_complex @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y2 )
        @ ( groups2073611262835488442omplex
          @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X2 @ P4 ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ N2 @ P4 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7327_diff__power__eq__sum,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N2 ) ) @ ( power_power_int @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X2 @ Y2 )
        @ ( groups3539618377306564664at_int
          @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X2 @ P4 ) @ ( power_power_int @ Y2 @ ( minus_minus_nat @ N2 @ P4 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7328_diff__power__eq__sum,axiom,
    ! [X2: real,N2: nat,Y2: real] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N2 ) ) @ ( power_power_real @ Y2 @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X2 @ Y2 )
        @ ( groups6591440286371151544t_real
          @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X2 @ P4 ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ N2 @ P4 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7329_power__diff__sumr2,axiom,
    ! [X2: complex,N2: nat,Y2: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y2 )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X2 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7330_power__diff__sumr2,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X2 @ Y2 )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X2 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7331_power__diff__sumr2,axiom,
    ! [X2: real,N2: nat,Y2: real] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X2 @ Y2 )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X2 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7332_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_7333_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_7334_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_7335_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_7336_one__diff__power__eq_H,axiom,
    ! [X2: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7337_one__diff__power__eq_H,axiom,
    ! [X2: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7338_one__diff__power__eq_H,axiom,
    ! [X2: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7339_one__diff__power__eq_H,axiom,
    ! [X2: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7340_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_7341_of__int__round__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7342_of__int__round__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7343_of__int__round__ge,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).

% of_int_round_ge
thf(fact_7344_of__int__round__ge,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).

% of_int_round_ge
thf(fact_7345_of__int__round__gt,axiom,
    ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).

% of_int_round_gt
thf(fact_7346_of__int__round__gt,axiom,
    ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).

% of_int_round_gt
thf(fact_7347_of__int__round__abs__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7348_of__int__round__abs__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7349_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_7350_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_7351_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7352_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7353_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_7354_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_7355_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7356_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7357_sum__less__suminf2,axiom,
    ! [F: nat > int,N2: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
       => ( ( 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_7358_sum__less__suminf2,axiom,
    ! [F: nat > nat,N2: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
       => ( ( 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_7359_sum__less__suminf2,axiom,
    ! [F: nat > real,N2: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
       => ( ( 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_7360_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_7361_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_7362_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7363_summable__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) ) ) ).

% summable_mult
thf(fact_7364_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) ) ) ).

% summable_mult
thf(fact_7365_summable__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ).

% summable_mult2
thf(fact_7366_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ).

% summable_mult2
thf(fact_7367_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7368_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7369_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7370_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) ) ) ).

% summable_divide
thf(fact_7371_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) ) ) ).

% summable_divide
thf(fact_7372_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7373_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7374_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7375_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7376_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7377_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7378_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7379_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7380_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7381_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7382_suminf__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
        = ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7383_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7384_suminf__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
        = ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).

% suminf_mult
thf(fact_7385_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_7386_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
            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7387_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
            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7388_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
            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7389_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7390_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7391_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7392_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7393_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7394_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7395_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7396_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7397_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7398_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7399_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7400_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_zero_power'
thf(fact_7401_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_zero_power'
thf(fact_7402_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).

% summable_zero_power'
thf(fact_7403_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_0_powser
thf(fact_7404_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_0_powser
thf(fact_7405_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7406_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7407_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7408_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7409_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7410_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7411_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7412_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7413_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs
thf(fact_7414_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7415_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7416_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7417_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7418_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7419_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7420_suminf__le__const,axiom,
    ! [F: nat > int,X2: int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X2 )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X2 ) ) ) ).

% suminf_le_const
thf(fact_7421_suminf__le__const,axiom,
    ! [F: nat > nat,X2: nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X2 )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X2 ) ) ) ).

% suminf_le_const
thf(fact_7422_suminf__le__const,axiom,
    ! [F: nat > real,X2: real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X2 )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X2 ) ) ) ).

% suminf_le_const
thf(fact_7423_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7424_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7425_summable__comparison__test_H,axiom,
    ! [G: nat > real,N3: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7426_summable__comparison__test_H,axiom,
    ! [G: nat > real,N3: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7427_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X2: int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X2 )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7428_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X2 )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7429_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X2: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X2 )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7430_complete__algebra__summable__geometric,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X2 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7431_complete__algebra__summable__geometric,axiom,
    ! [X2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7432_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_7433_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_7434_suminf__split__head,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7435_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7436_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) ) ).

% summable_norm
thf(fact_7437_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm
thf(fact_7438_sum__le__suminf,axiom,
    ! [F: nat > int,I6: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N4: nat] :
              ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7439_sum__le__suminf,axiom,
    ! [F: nat > nat,I6: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N4: nat] :
              ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7440_sum__le__suminf,axiom,
    ! [F: nat > real,I6: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N4: nat] :
              ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7441_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7442_suminf__minus__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7443_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7444_sum__less__suminf,axiom,
    ! [F: nat > int,N2: nat] :
      ( ( summable_int @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7445_sum__less__suminf,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7446_sum__less__suminf,axiom,
    ! [F: nat > real,N2: nat] :
      ( ( summable_real @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7447_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7448_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7449_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7450_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7451_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M2 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7452_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M2 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7453_suminf__exist__split,axiom,
    ! [R: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R )
     => ( ( 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 ) ) ) )
              @ R ) ) ) ) ).

% suminf_exist_split
thf(fact_7454_suminf__exist__split,axiom,
    ! [R: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R )
     => ( ( 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 ) ) ) )
              @ R ) ) ) ) ).

% suminf_exist_split
thf(fact_7455_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I4: nat] : ( ord_less_eq_real @ ( F @ I4 ) @ one_one_real )
     => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
       => ( ( 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_7456_Abel__lemma,axiom,
    ! [R: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R )
     => ( ( ord_less_real @ R @ R0 )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R0 @ N4 ) ) @ M7 )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R @ N ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_7457_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_7458_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_7459_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_7460_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_7461_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) )
        @ ( 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_7462_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_7463_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X5: nat > real] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_real @ ( X5 @ M6 ) @ ( X5 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7464_monoseq__def,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X5: nat > set_nat] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_set_nat @ ( X5 @ M6 ) @ ( X5 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_set_nat @ ( X5 @ N ) @ ( X5 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7465_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X5: nat > rat] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_rat @ ( X5 @ M6 ) @ ( X5 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7466_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X5: nat > num] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_num @ ( X5 @ M6 ) @ ( X5 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7467_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X5: nat > nat] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_nat @ ( X5 @ M6 ) @ ( X5 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7468_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X5: nat > int] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_int @ ( X5 @ M6 ) @ ( X5 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7469_monoI2,axiom,
    ! [X8: nat > real] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_real @ ( X8 @ N4 ) @ ( X8 @ M5 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% monoI2
thf(fact_7470_monoI2,axiom,
    ! [X8: nat > set_nat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_set_nat @ ( X8 @ N4 ) @ ( X8 @ M5 ) ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% monoI2
thf(fact_7471_monoI2,axiom,
    ! [X8: nat > rat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( X8 @ M5 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% monoI2
thf(fact_7472_monoI2,axiom,
    ! [X8: nat > num] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_num @ ( X8 @ N4 ) @ ( X8 @ M5 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% monoI2
thf(fact_7473_monoI2,axiom,
    ! [X8: nat > nat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_nat @ ( X8 @ N4 ) @ ( X8 @ M5 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% monoI2
thf(fact_7474_monoI2,axiom,
    ! [X8: nat > int] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_int @ ( X8 @ N4 ) @ ( X8 @ M5 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% monoI2
thf(fact_7475_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_7476_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X2: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
        @ X2 )
      = ( ( A @ zero_zero_nat )
        = X2 ) ) ).

% powser_sums_zero_iff
thf(fact_7477_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X2: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
        @ X2 )
      = ( ( A @ zero_zero_nat )
        = X2 ) ) ).

% powser_sums_zero_iff
thf(fact_7478_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S3: real,T: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( sums_real @ F @ S3 )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S3 @ T ) ) ) ) ).

% sums_le
thf(fact_7479_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S3: nat,T: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( sums_nat @ F @ S3 )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S3 @ T ) ) ) ) ).

% sums_le
thf(fact_7480_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S3: int,T: int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( sums_int @ F @ S3 )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S3 @ T ) ) ) ) ).

% sums_le
thf(fact_7481_sums__mult2,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
        @ ( times_times_complex @ A @ C ) ) ) ).

% sums_mult2
thf(fact_7482_sums__mult2,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
        @ ( times_times_real @ A @ C ) ) ) ).

% sums_mult2
thf(fact_7483_sums__mult,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
        @ ( times_times_complex @ C @ A ) ) ) ).

% sums_mult
thf(fact_7484_sums__mult,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
        @ ( times_times_real @ C @ A ) ) ) ).

% sums_mult
thf(fact_7485_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_7486_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_7487_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_7488_sums__divide,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C )
        @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).

% sums_divide
thf(fact_7489_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_7490_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_7491_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_7492_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_7493_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_7494_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7495_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7496_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7497_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_7498_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_7499_sums__Suc__imp,axiom,
    ! [F: nat > complex,S3: complex] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S3 )
       => ( sums_complex @ F @ S3 ) ) ) ).

% sums_Suc_imp
thf(fact_7500_sums__Suc__imp,axiom,
    ! [F: nat > real,S3: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S3 )
       => ( sums_real @ F @ S3 ) ) ) ).

% sums_Suc_imp
thf(fact_7501_sums__Suc,axiom,
    ! [F: nat > real,L2: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L2 )
     => ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_7502_sums__Suc,axiom,
    ! [F: nat > nat,L2: nat] :
      ( ( sums_nat
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L2 )
     => ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_7503_sums__Suc,axiom,
    ! [F: nat > int,L2: int] :
      ( ( sums_int
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L2 )
     => ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_7504_sums__Suc__iff,axiom,
    ! [F: nat > real,S3: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ S3 )
      = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_7505_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > complex,S3: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ( F @ I4 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S3 )
        = ( sums_complex @ F @ S3 ) ) ) ).

% sums_zero_iff_shift
thf(fact_7506_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S3: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ( F @ I4 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S3 )
        = ( sums_real @ F @ S3 ) ) ) ).

% sums_zero_iff_shift
thf(fact_7507_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_7508_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_7509_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_7510_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_7511_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_7512_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S3: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ S3 )
      = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_7513_sums__iff__shift_H,axiom,
    ! [F: nat > complex,N2: nat,S3: complex] :
      ( ( sums_complex
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_complex @ F @ S3 ) ) ).

% sums_iff_shift'
thf(fact_7514_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S3: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S3 ) ) ).

% sums_iff_shift'
thf(fact_7515_sums__split__initial__segment,axiom,
    ! [F: nat > complex,S3: complex,N2: nat] :
      ( ( sums_complex @ F @ S3 )
     => ( sums_complex
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_7516_sums__split__initial__segment,axiom,
    ! [F: nat > real,S3: real,N2: nat] :
      ( ( sums_real @ F @ S3 )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_7517_sums__If__finite__set_H,axiom,
    ! [G: nat > complex,S: complex,A2: set_nat,S4: complex,F: nat > complex] :
      ( ( sums_complex @ G @ S )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S4
            = ( plus_plus_complex @ S
              @ ( groups2073611262835488442omplex
                @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) )
                @ A2 ) ) )
         => ( sums_complex
            @ ^ [N: nat] : ( if_complex @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
            @ S4 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_7518_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S: real,A2: set_nat,S4: real,F: nat > real] :
      ( ( sums_real @ G @ S )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S4
            = ( plus_plus_real @ S
              @ ( groups6591440286371151544t_real
                @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
                @ A2 ) ) )
         => ( sums_real
            @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
            @ S4 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_7519_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_7520_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7521_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_7522_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_7523_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_7524_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_7525_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_7526_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_7527_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_7528_arctan__ubound,axiom,
    ! [Y2: real] : ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_7529_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_7530_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_7531_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_7532_power__half__series,axiom,
    ( sums_real
    @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_7533_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_7534_arctan__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) )
      & ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_7535_arctan__lbound,axiom,
    ! [Y2: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) ) ).

% arctan_lbound
thf(fact_7536_sums__if_H,axiom,
    ! [G: nat > real,X2: real] :
      ( ( sums_real @ G @ X2 )
     => ( sums_real
        @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X2 ) ) ).

% sums_if'
thf(fact_7537_sums__if,axiom,
    ! [G: nat > real,X2: real,F: nat > real,Y2: real] :
      ( ( sums_real @ G @ X2 )
     => ( ( sums_real @ F @ Y2 )
       => ( sums_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( F @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X2 @ Y2 ) ) ) ) ).

% sums_if
thf(fact_7538_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_7539_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_7540_mono__SucI1,axiom,
    ! [X8: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% mono_SucI1
thf(fact_7541_mono__SucI1,axiom,
    ! [X8: nat > set_nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% mono_SucI1
thf(fact_7542_mono__SucI1,axiom,
    ! [X8: nat > rat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% mono_SucI1
thf(fact_7543_mono__SucI1,axiom,
    ! [X8: nat > num] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% mono_SucI1
thf(fact_7544_mono__SucI1,axiom,
    ! [X8: nat > nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% mono_SucI1
thf(fact_7545_mono__SucI1,axiom,
    ! [X8: nat > int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% mono_SucI1
thf(fact_7546_mono__SucI2,axiom,
    ! [X8: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% mono_SucI2
thf(fact_7547_mono__SucI2,axiom,
    ! [X8: nat > set_nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% mono_SucI2
thf(fact_7548_mono__SucI2,axiom,
    ! [X8: nat > rat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% mono_SucI2
thf(fact_7549_mono__SucI2,axiom,
    ! [X8: nat > num] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% mono_SucI2
thf(fact_7550_mono__SucI2,axiom,
    ! [X8: nat > nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% mono_SucI2
thf(fact_7551_mono__SucI2,axiom,
    ! [X8: nat > int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% mono_SucI2
thf(fact_7552_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X5: nat > real] :
          ( ! [N: nat] : ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_real @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7553_monoseq__Suc,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X5: nat > set_nat] :
          ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_set_nat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7554_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X5: nat > rat] :
          ( ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7555_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X5: nat > num] :
          ( ! [N: nat] : ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_num @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7556_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X5: nat > nat] :
          ( ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7557_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X5: nat > int] :
          ( ! [N: nat] : ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7558_monoI1,axiom,
    ! [X8: nat > real] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% monoI1
thf(fact_7559_monoI1,axiom,
    ! [X8: nat > set_nat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_set_nat @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) )
     => ( topolo7278393974255667507et_nat @ X8 ) ) ).

% monoI1
thf(fact_7560_monoI1,axiom,
    ! [X8: nat > rat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_rat @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% monoI1
thf(fact_7561_monoI1,axiom,
    ! [X8: nat > num] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_num @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% monoI1
thf(fact_7562_monoI1,axiom,
    ! [X8: nat > nat] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_nat @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% monoI1
thf(fact_7563_monoI1,axiom,
    ! [X8: nat > int] :
      ( ! [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
         => ( ord_less_eq_int @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% monoI1
thf(fact_7564_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_7565_diffs__equiv,axiom,
    ! [C: nat > complex,X2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X2 @ N ) ) )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X2 @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_7566_diffs__equiv,axiom,
    ! [C: nat > real,X2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X2 @ N ) ) )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_7567_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_7568_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_7569_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_7570_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_7571_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
          @ N
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_7572_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_7573_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_7574_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_7575_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_7576_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_7577_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_7578_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_7579_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_7580_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_7581_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_7582_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_7583_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_7584_cos__periodic__pi,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).

% cos_periodic_pi
thf(fact_7585_cos__periodic__pi2,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X2 ) )
      = ( uminus_uminus_real @ ( cos_real @ X2 ) ) ) ).

% cos_periodic_pi2
thf(fact_7586_sin__periodic__pi,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_periodic_pi
thf(fact_7587_sin__periodic__pi2,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X2 ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_periodic_pi2
thf(fact_7588_sin__cos__squared__add3,axiom,
    ! [X2: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ X2 ) ) @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( sin_complex @ X2 ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_7589_sin__cos__squared__add3,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ X2 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ X2 ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_7590_sin__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_7591_sin__npi2,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_7592_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_7593_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_7594_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_7595_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_7596_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_7597_cos__periodic,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X2 ) ) ).

% cos_periodic
thf(fact_7598_sin__periodic,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X2 ) ) ).

% sin_periodic
thf(fact_7599_cos__2pi__minus,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
      = ( cos_real @ X2 ) ) ).

% cos_2pi_minus
thf(fact_7600_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_7601_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_7602_sin__cos__squared__add2,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_7603_sin__cos__squared__add2,axiom,
    ! [X2: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_7604_sin__cos__squared__add,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_7605_sin__cos__squared__add,axiom,
    ! [X2: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_7606_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_7607_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_7608_sin__2pi__minus,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_2pi_minus
thf(fact_7609_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_7610_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_7611_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_7612_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_7613_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_7614_cos__one__sin__zero,axiom,
    ! [X2: complex] :
      ( ( ( cos_complex @ X2 )
        = one_one_complex )
     => ( ( sin_complex @ X2 )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_7615_cos__one__sin__zero,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = one_one_real )
     => ( ( sin_real @ X2 )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_7616_polar__Ex,axiom,
    ! [X2: real,Y2: real] :
    ? [R3: real,A3: real] :
      ( ( X2
        = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
      & ( Y2
        = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_7617_sin__diff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( sin_complex @ ( minus_minus_complex @ X2 @ Y2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( cos_complex @ Y2 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( sin_complex @ Y2 ) ) ) ) ).

% sin_diff
thf(fact_7618_sin__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( sin_real @ ( minus_minus_real @ X2 @ Y2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( sin_real @ Y2 ) ) ) ) ).

% sin_diff
thf(fact_7619_sin__add,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( sin_complex @ ( plus_plus_complex @ X2 @ Y2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( cos_complex @ Y2 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( sin_complex @ Y2 ) ) ) ) ).

% sin_add
thf(fact_7620_sin__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( sin_real @ Y2 ) ) ) ) ).

% sin_add
thf(fact_7621_cos__diff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X2 @ Y2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y2 ) ) @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( sin_complex @ Y2 ) ) ) ) ).

% cos_diff
thf(fact_7622_cos__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( cos_real @ ( minus_minus_real @ X2 @ Y2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) ) ) ) ).

% cos_diff
thf(fact_7623_cos__add,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X2 @ Y2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y2 ) ) @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( sin_complex @ Y2 ) ) ) ) ).

% cos_add
thf(fact_7624_cos__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) ) ) ) ).

% cos_add
thf(fact_7625_sin__zero__norm__cos__one,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X2 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7626_sin__zero__norm__cos__one,axiom,
    ! [X2: complex] :
      ( ( ( sin_complex @ X2 )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X2 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7627_sin__zero__abs__cos__one,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X2 ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_7628_sin__double,axiom,
    ! [X2: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X2 ) ) @ ( cos_complex @ X2 ) ) ) ).

% sin_double
thf(fact_7629_sin__double,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X2 ) ) @ ( cos_real @ X2 ) ) ) ).

% sin_double
thf(fact_7630_sincos__principal__value,axiom,
    ! [X2: real] :
    ? [Y5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y5 )
      & ( ord_less_eq_real @ Y5 @ pi )
      & ( ( sin_real @ Y5 )
        = ( sin_real @ X2 ) )
      & ( ( cos_real @ Y5 )
        = ( cos_real @ X2 ) ) ) ).

% sincos_principal_value
thf(fact_7631_sin__x__le__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).

% sin_x_le_x
thf(fact_7632_sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).

% sin_le_one
thf(fact_7633_cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).

% cos_le_one
thf(fact_7634_abs__sin__x__le__abs__x,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_7635_pochhammer__pos,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7636_pochhammer__pos,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7637_pochhammer__pos,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X2 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7638_pochhammer__pos,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_7639_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_7640_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_7641_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_7642_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_7643_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_7644_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_7645_sin__cos__le1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_7646_sin__squared__eq,axiom,
    ! [X2: complex] :
      ( ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7647_sin__squared__eq,axiom,
    ! [X2: real] :
      ( ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7648_cos__squared__eq,axiom,
    ! [X2: complex] :
      ( ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7649_cos__squared__eq,axiom,
    ! [X2: real] :
      ( ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7650_sin__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero
thf(fact_7651_sin__x__ge__neg__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_7652_sin__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_ge_zero
thf(fact_7653_sin__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).

% sin_ge_minus_one
thf(fact_7654_cos__inj__pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ pi )
           => ( ( ( cos_real @ X2 )
                = ( cos_real @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_7655_cos__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) )
              = ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_7656_cos__monotone__0__pi__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_7657_cos__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).

% cos_ge_minus_one
thf(fact_7658_abs__sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_7659_abs__cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_7660_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_7661_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_7662_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_7663_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_7664_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_7665_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_7666_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_7667_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_7668_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_7669_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_7670_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_7671_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_7672_cos__double,axiom,
    ! [X2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7673_cos__double,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7674_pochhammer__nonneg,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7675_pochhammer__nonneg,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7676_pochhammer__nonneg,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7677_pochhammer__nonneg,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X2 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_7678_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_7679_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_7680_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_7681_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_7682_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_7683_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_7684_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_7685_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_7686_cos__monotone__0__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_7687_cos__mono__less__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ pi )
           => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) )
              = ( ord_less_real @ Y2 @ X2 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_7688_sin__eq__0__pi,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
     => ( ( ord_less_real @ X2 @ pi )
       => ( ( ( sin_real @ X2 )
            = zero_zero_real )
         => ( X2 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_7689_sin__zero__pi__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ pi )
     => ( ( ( sin_real @ X2 )
          = zero_zero_real )
        = ( X2 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_7690_cos__monotone__minus__pi__0_H,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y2 ) @ ( cos_real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_7691_sin__zero__iff__int2,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( X2
            = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_7692_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_7693_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_7694_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_7695_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_7696_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_7697_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C2: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7698_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C2: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7699_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C2: nat > rat,N: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7700_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C2: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7701_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_7702_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_7703_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_7704_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_7705_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_7706_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_7707_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_7708_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_7709_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_7710_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_7711_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_7712_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_7713_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_7714_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_7715_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_7716_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_7717_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_7718_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_7719_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_7720_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_7721_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7722_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7723_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7724_sincos__total__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ pi )
            & ( X2
              = ( cos_real @ T4 ) )
            & ( Y2
              = ( sin_real @ T4 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_7725_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_7726_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_7727_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_7728_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_7729_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_7730_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_7731_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_7732_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_7733_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_7734_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_7735_sin__expansion__lemma,axiom,
    ! [X2: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7736_cos__expansion__lemma,axiom,
    ! [X2: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7737_termdiff__converges__all,axiom,
    ! [C: nat > complex,X2: complex] :
      ( ! [X3: complex] :
          ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X3 @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X2 @ N ) ) ) ) ).

% termdiff_converges_all
thf(fact_7738_termdiff__converges__all,axiom,
    ! [C: nat > real,X2: real] :
      ( ! [X3: real] :
          ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X3 @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X2 @ N ) ) ) ) ).

% termdiff_converges_all
thf(fact_7739_sin__gt__zero__02,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_7740_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_7741_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 )
      & ! [Y3: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
            & ( ord_less_eq_real @ Y3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y3 )
              = zero_zero_real ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
thf(fact_7742_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_7743_cos__monotone__minus__pi__0,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y2 ) @ ( cos_real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_7744_cos__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y2 )
            & ! [Y3: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
                  & ( ord_less_eq_real @ Y3 @ pi )
                  & ( ( cos_real @ Y3 )
                    = Y2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_7745_sincos__total__pi__half,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T4: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X2
                = ( cos_real @ T4 ) )
              & ( Y2
                = ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_7746_sincos__total__2pi__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X2
            = ( cos_real @ T4 ) )
          & ( Y2
            = ( sin_real @ T4 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_7747_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_7748_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_7749_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_7750_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_7751_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_7752_sincos__total__2pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X2
                  = ( cos_real @ T4 ) )
               => ( Y2
                 != ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_7753_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_7754_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
      = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7755_pochhammer__absorb__comp,axiom,
    ! [R: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
      = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7756_pochhammer__absorb__comp,axiom,
    ! [R: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
      = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7757_pochhammer__absorb__comp,axiom,
    ! [R: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
      = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7758_pochhammer__absorb__comp,axiom,
    ! [R: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
      = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7759_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_7760_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_7761_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_7762_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_7763_sin__gt__zero2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero2
thf(fact_7764_sin__lt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ pi @ X2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_7765_cos__double__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_7766_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_7767_cos__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_gt_zero
thf(fact_7768_sin__monotone__2pi__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y2 ) @ ( sin_real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_7769_sin__mono__le__eq,axiom,
    ! [X2: real,Y2: 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 ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) )
              = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_7770_sin__inj__pi,axiom,
    ! [X2: real,Y2: 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 ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X2 )
                = ( sin_real @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_7771_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_7772_cos__one__2pi__int,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = one_one_real )
      = ( ? [X: int] :
            ( X2
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_7773_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_7774_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_7775_cos__treble__cos,axiom,
    ! [X2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X2 ) ) ) ) ).

% cos_treble_cos
thf(fact_7776_cos__treble__cos,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X2 ) ) ) ) ).

% cos_treble_cos
thf(fact_7777_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_7778_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_7779_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_7780_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_7781_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_7782_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_7783_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_7784_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_7785_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_7786_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_7787_termdiff__converges,axiom,
    ! [X2: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K5 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
           => ( summable_real
              @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X3 @ N ) ) ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).

% termdiff_converges
thf(fact_7788_termdiff__converges,axiom,
    ! [X2: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K5 )
     => ( ! [X3: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
           => ( summable_complex
              @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X2 @ N ) ) ) ) ) ).

% termdiff_converges
thf(fact_7789_sin__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ pi @ X2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_7790_sin__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_7791_sin__monotone__2pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y2 ) @ ( sin_real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_7792_sin__mono__less__eq,axiom,
    ! [X2: real,Y2: 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 ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) )
              = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_7793_sin__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ 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 )
              = Y2 )
            & ! [Y3: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
                  & ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y3 )
                    = Y2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_7794_cos__gt__zero__pi,axiom,
    ! [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 ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_7795_cos__ge__zero,axiom,
    ! [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 ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_ge_zero
thf(fact_7796_cos__one__2pi,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = one_one_real )
      = ( ? [X: nat] :
            ( X2
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X: nat] :
            ( X2
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_7797_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_7798_sin__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X2
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_7799_cos__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X2
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_7800_sin__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( sin_real @ X2 )
          = zero_zero_real )
       => ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_7801_sin__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_7802_cos__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( cos_real @ X2 )
          = zero_zero_real )
       => ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_7803_cos__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = zero_zero_real )
      = ( ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_7804_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ? [T4: real] :
            ( ( ord_less_real @ X2 @ T4 )
            & ( ord_less_real @ T4 @ zero_zero_real )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_7805_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X2 )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_7806_Maclaurin__cos__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
      & ( ( cos_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_7807_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
        @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( 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_7808_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
        @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( 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_7809_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
        @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( 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_7810_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A4: complex,N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_7811_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_7812_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_7813_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_7814_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_7815_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_7816_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7817_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7818_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_7819_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_7820_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_7821_prod_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
      = one_one_complex ) ).

% prod.empty
thf(fact_7822_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_7823_prod_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
      = one_one_rat ) ).

% prod.empty
thf(fact_7824_prod_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
      = one_one_nat ) ).

% prod.empty
thf(fact_7825_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_7826_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_7827_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_7828_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7829_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7830_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7831_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7832_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7833_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7834_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_7835_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.infinite
thf(fact_7836_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups708209901874060359at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_7837_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups705719431365010083at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.infinite
thf(fact_7838_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_7839_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_7840_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_7841_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_7842_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_7843_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_7844_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_7845_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_7846_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_7847_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_7848_prod_Odelta_H,axiom,
    ! [S: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A @ S )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7849_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7850_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7851_prod_Odelta_H,axiom,
    ! [S: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A @ S )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7852_prod_Odelta_H,axiom,
    ! [S: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A @ S )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7853_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7854_prod_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7855_prod_Odelta_H,axiom,
    ! [S: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A @ S )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7856_prod_Odelta_H,axiom,
    ! [S: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A @ S )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_7857_prod_Odelta_H,axiom,
    ! [S: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_7858_prod_Odelta,axiom,
    ! [S: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A @ S )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7859_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7860_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7861_prod_Odelta,axiom,
    ! [S: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A @ S )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7862_prod_Odelta,axiom,
    ! [S: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A @ S )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7863_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7864_prod_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7865_prod_Odelta,axiom,
    ! [S: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A @ S )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7866_prod_Odelta,axiom,
    ! [S: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A @ S )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_7867_prod_Odelta,axiom,
    ! [S: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A @ S )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_7868_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_7869_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_7870_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7871_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7872_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7873_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_7874_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_Suc
thf(fact_7875_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_7876_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_7877_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_7878_prod_OlessThan__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_7879_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_7880_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_7881_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_7882_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7883_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7884_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7885_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7886_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7887_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_7888_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_7889_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_7890_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_7891_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_7892_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7893_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups713298508707869441omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7894_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups7440179247065528705omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7895_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > complex,A2: set_complex] :
      ( ( ( groups3708469109370488835omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7896_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A2: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7897_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7898_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7899_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A2: set_complex] :
      ( ( ( groups766887009212190081x_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7900_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7901_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups4061424788464935467al_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7902_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_7903_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_7904_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_7905_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_7906_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_7907_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_7908_prod__power__distrib,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_7909_prod__power__distrib,axiom,
    ! [F: nat > int,A2: set_nat,N2: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_7910_prod__power__distrib,axiom,
    ! [F: int > int,A2: set_int,N2: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_7911_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_7912_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_7913_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_7914_fact__prod,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N: nat] :
          ( semiri8010041392384452111omplex
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_7915_fact__prod,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N: nat] :
          ( semiri681578069525770553at_rat
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_7916_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_7917_fact__prod,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N: nat] :
          ( semiri5074537144036343181t_real
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_7918_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).

% fact_prod
thf(fact_7919_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_7920_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_7921_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_7922_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7923_prod__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7924_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7925_prod__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
            & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7926_prod__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
            & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7927_prod__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
            & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7928_prod__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
            & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7929_prod__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
            & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7930_prod__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
            & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7931_prod__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
            & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_7932_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_7933_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_7934_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_7935_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_7936_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_7937_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_7938_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_7939_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_7940_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_7941_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_7942_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_7943_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_7944_prod__ge__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_7945_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7946_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_7947_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_7948_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7949_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_7950_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_7951_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_7952_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_7953_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_7954_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_7955_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_7956_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_7957_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_7958_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_7959_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7960_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7961_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7962_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_7963_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_7964_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_7965_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_7966_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_7967_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_7968_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_7969_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_7970_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_7971_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_7972_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_7973_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_7974_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups7440179247065528705omplex
          @ ^ [X: int] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7975_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups713298508707869441omplex @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups713298508707869441omplex
          @ ^ [X: real] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7976_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups6464643781859351333omplex
          @ ^ [X: nat] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7977_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups3708469109370488835omplex
          @ ^ [X: complex] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ one_one_complex )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7978_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7979_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups1681761925125756287l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7980_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X: nat] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7981_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups766887009212190081x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7982_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups1072433553688619179nt_rat
          @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7983_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups4061424788464935467al_rat
          @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_7984_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_7985_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_7986_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_7987_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_7988_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_7989_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_7990_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_7991_power__sum,axiom,
    ! [C: real,F: nat > nat,A2: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [A4: nat] : ( power_power_real @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_7992_power__sum,axiom,
    ! [C: complex,F: nat > nat,A2: set_nat] :
      ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [A4: nat] : ( power_power_complex @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_7993_power__sum,axiom,
    ! [C: nat,F: nat > nat,A2: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A4: nat] : ( power_power_nat @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_7994_power__sum,axiom,
    ! [C: int,F: nat > nat,A2: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [A4: nat] : ( power_power_int @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_7995_power__sum,axiom,
    ! [C: int,F: int > nat,A2: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A4: int] : ( power_power_int @ C @ ( F @ A4 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_7996_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_7997_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_7998_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_7999_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_8000_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_8001_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_8002_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_8003_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_8004_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_8005_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_8006_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_8007_prod__le__1,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 @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8008_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8009_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S ) @ ( groups225925009352817453ex_rat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8010_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups6464643781859351333omplex @ H2 @ S ) @ ( groups6464643781859351333omplex @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8011_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups3708469109370488835omplex @ H2 @ S ) @ ( groups3708469109370488835omplex @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8012_prod_Orelated,axiom,
    ! [R2: real > real > $o,S: set_nat,H2: nat > real,G: nat > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups129246275422532515t_real @ H2 @ S ) @ ( groups129246275422532515t_real @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8013_prod_Orelated,axiom,
    ! [R2: real > real > $o,S: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups766887009212190081x_real @ H2 @ S ) @ ( groups766887009212190081x_real @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8014_prod_Orelated,axiom,
    ! [R2: nat > nat > $o,S: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R2 @ one_one_nat @ one_one_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups861055069439313189ex_nat @ H2 @ S ) @ ( groups861055069439313189ex_nat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8015_prod_Orelated,axiom,
    ! [R2: int > int > $o,S: set_complex,H2: complex > int,G: complex > int] :
      ( ( R2 @ one_one_int @ one_one_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_int @ X15 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups858564598930262913ex_int @ H2 @ S ) @ ( groups858564598930262913ex_int @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8016_prod_Orelated,axiom,
    ! [R2: nat > nat > $o,S: set_nat,H2: nat > nat,G: nat > nat] :
      ( ( R2 @ one_one_nat @ one_one_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups708209901874060359at_nat @ H2 @ S ) @ ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8017_prod_Orelated,axiom,
    ! [R2: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
      ( ( R2 @ one_one_int @ one_one_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_int @ X15 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R2 @ ( groups705719431365010083at_int @ H2 @ S ) @ ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ) ).

% prod.related
thf(fact_8018_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8019_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8020_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8021_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8022_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8023_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8024_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A2 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8025_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8026_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A2 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8027_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8028_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8029_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8030_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8031_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8032_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8033_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8034_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: int > int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8035_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_real,S: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S )
                        = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8036_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_int,S: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S )
                        = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8037_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_real,S: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S )
                        = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8038_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_int,S: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S )
                        = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8039_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_complex,S: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > complex,H2: complex > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S )
                        = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8040_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_complex,S: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > complex,H2: complex > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S )
                        = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8041_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T5: set_real,S: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > complex,H2: real > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S )
                        = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8042_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T5: set_int,S: set_complex,I: int > complex,J: complex > int,T3: set_int,G: complex > complex,H2: int > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S )
                        = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8043_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T5: set_complex,S: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S )
                        = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8044_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_real,S: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T5 )
                       => ( ( H2 @ B3 )
                          = one_one_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1681761925125756287l_real @ G @ S )
                        = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_8045_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_8046_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_8047_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_8048_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_8049_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_8050_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_8051_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_8052_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_8053_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_8054_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_8055_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_8056_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_8057_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_8058_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_8059_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > complex] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups713298508707869441omplex @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = one_one_complex ) ) ) )
        = ( groups713298508707869441omplex @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8060_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = one_one_complex ) ) ) )
        = ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8061_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = one_one_real ) ) ) )
        = ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8062_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = one_one_real ) ) ) )
        = ( groups766887009212190081x_real @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8063_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = one_one_rat ) ) ) )
        = ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8064_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = one_one_rat ) ) ) )
        = ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8065_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4696554848551431203al_nat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = one_one_nat ) ) ) )
        = ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8066_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = one_one_nat ) ) ) )
        = ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8067_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4694064378042380927al_int @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = one_one_int ) ) ) )
        = ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8068_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = one_one_int ) ) ) )
        = ( groups858564598930262913ex_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_8069_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8070_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8071_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8072_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8073_fact__code,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8074_fact__code,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8075_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8076_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8077_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8078_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8079_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8080_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: nat] :
                ( ( member_nat @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8081_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8082_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8083_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I4: int] :
                ( ( member_int @ I4 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8084_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I4: nat] :
                ( ( member_nat @ I4 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8085_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8086_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: real] :
                ( ( member_real @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8087_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I4: complex] :
                ( ( member_complex @ I4 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8088_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8089_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8090_less__1__prod,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8091_less__1__prod,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8092_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8093_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8094_less__1__prod,axiom,
    ! [I6: set_int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8095_less__1__prod,axiom,
    ! [I6: set_real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8096_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8097_less__1__prod,axiom,
    ! [I6: set_real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_8098_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > complex] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups3708469109370488835omplex @ G @ A2 )
          = ( times_times_complex @ ( groups3708469109370488835omplex @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups3708469109370488835omplex @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8099_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups766887009212190081x_real @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8100_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ A2 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups861055069439313189ex_nat @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8101_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ A2 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups858564598930262913ex_int @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8102_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > complex] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups6464643781859351333omplex @ G @ A2 )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups6464643781859351333omplex @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8103_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups129246275422532515t_real @ G @ A2 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups129246275422532515t_real @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8104_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups708209901874060359at_nat @ G @ A2 )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups708209901874060359at_nat @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8105_prod_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups705719431365010083at_int @ G @ A2 )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups705719431365010083at_int @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8106_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups1705073143266064639nt_int @ G @ A2 )
          = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1705073143266064639nt_int @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8107_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups713298508707869441omplex @ G @ T3 )
              = ( groups713298508707869441omplex @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8108_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ T3 )
              = ( groups7440179247065528705omplex @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8109_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ T3 )
              = ( groups3708469109370488835omplex @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8110_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T3 )
              = ( groups1681761925125756287l_real @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8111_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ T3 )
              = ( groups2316167850115554303t_real @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8112_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups766887009212190081x_real @ G @ T3 )
              = ( groups766887009212190081x_real @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8113_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ T3 )
              = ( groups4061424788464935467al_rat @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8114_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ T3 )
              = ( groups1072433553688619179nt_rat @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8115_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups225925009352817453ex_rat @ G @ T3 )
              = ( groups225925009352817453ex_rat @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8116_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ T3 )
              = ( groups4696554848551431203al_nat @ H2 @ S ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8117_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups713298508707869441omplex @ G @ S )
              = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8118_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ S )
              = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8119_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_complex ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ S )
              = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8120_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S )
              = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8121_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ S )
              = ( groups2316167850115554303t_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8122_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups766887009212190081x_real @ G @ S )
              = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8123_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ S )
              = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8124_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S @ T3 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ S )
              = ( groups1072433553688619179nt_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8125_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups225925009352817453ex_rat @ G @ S )
              = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8126_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S @ T3 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S ) )
             => ( ( H2 @ X3 )
                = one_one_nat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ S )
              = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8127_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ T3 )
            = ( groups3708469109370488835omplex @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8128_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T3 )
            = ( groups766887009212190081x_real @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8129_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ T3 )
            = ( groups225925009352817453ex_rat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8130_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ T3 )
            = ( groups861055069439313189ex_nat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8131_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ T3 )
            = ( groups858564598930262913ex_int @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8132_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ T3 )
            = ( groups6464643781859351333omplex @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8133_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ T3 )
            = ( groups129246275422532515t_real @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8134_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ T3 )
            = ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8135_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups708209901874060359at_nat @ G @ T3 )
            = ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8136_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups705719431365010083at_int @ G @ T3 )
            = ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8137_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ S )
            = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8138_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S )
            = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8139_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ S )
            = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8140_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ S )
            = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8141_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S @ T3 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ S )
            = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8142_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ S )
            = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8143_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ S )
            = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8144_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ S )
            = ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8145_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups708209901874060359at_nat @ G @ S )
            = ( groups708209901874060359at_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8146_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S @ T3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups705719431365010083at_int @ G @ S )
            = ( groups705719431365010083at_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8147_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) )
               => ( ( groups713298508707869441omplex @ G @ A2 )
                  = ( groups713298508707869441omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8148_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ C4 )
                  = ( groups7440179247065528705omplex @ H2 @ C4 ) )
               => ( ( groups7440179247065528705omplex @ G @ A2 )
                  = ( groups7440179247065528705omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8149_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) )
               => ( ( groups3708469109370488835omplex @ G @ A2 )
                  = ( groups3708469109370488835omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8150_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8151_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ C4 )
                  = ( groups2316167850115554303t_real @ H2 @ C4 ) )
               => ( ( groups2316167850115554303t_real @ G @ A2 )
                  = ( groups2316167850115554303t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8152_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) )
               => ( ( groups766887009212190081x_real @ G @ A2 )
                  = ( groups766887009212190081x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8153_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
               => ( ( groups4061424788464935467al_rat @ G @ A2 )
                  = ( groups4061424788464935467al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8154_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ C4 ) )
               => ( ( groups1072433553688619179nt_rat @ G @ A2 )
                  = ( groups1072433553688619179nt_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8155_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G @ C4 )
                  = ( groups225925009352817453ex_rat @ H2 @ C4 ) )
               => ( ( groups225925009352817453ex_rat @ G @ A2 )
                  = ( groups225925009352817453ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8156_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) )
               => ( ( groups4696554848551431203al_nat @ G @ A2 )
                  = ( groups4696554848551431203al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8157_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ A2 )
                  = ( groups713298508707869441omplex @ H2 @ B2 ) )
                = ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8158_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ A2 )
                  = ( groups7440179247065528705omplex @ H2 @ B2 ) )
                = ( ( groups7440179247065528705omplex @ G @ C4 )
                  = ( groups7440179247065528705omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8159_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ A2 )
                  = ( groups3708469109370488835omplex @ H2 @ B2 ) )
                = ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8160_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B2 ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8161_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ A2 )
                  = ( groups2316167850115554303t_real @ H2 @ B2 ) )
                = ( ( groups2316167850115554303t_real @ G @ C4 )
                  = ( groups2316167850115554303t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8162_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ A2 )
                  = ( groups766887009212190081x_real @ H2 @ B2 ) )
                = ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8163_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ A2 )
                  = ( groups4061424788464935467al_rat @ H2 @ B2 ) )
                = ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8164_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
                  = ( groups1072433553688619179nt_rat @ H2 @ B2 ) )
                = ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8165_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G @ A2 )
                  = ( groups225925009352817453ex_rat @ H2 @ B2 ) )
                = ( ( groups225925009352817453ex_rat @ G @ C4 )
                  = ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8166_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ A2 )
                  = ( groups4696554848551431203al_nat @ H2 @ B2 ) )
                = ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8167_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8168_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8169_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8170_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8171_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_complex @ ( G @ ( suc @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8172_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8173_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8174_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8175_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( G @ M ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8176_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8177_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8178_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8179_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_8180_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_8181_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_8182_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_8183_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8184_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8185_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8186_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8187_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_complex @ ( G @ M )
          @ ( groups6464643781859351333omplex
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8188_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ M )
          @ ( groups129246275422532515t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8189_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8190_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8191_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8192_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_8193_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_8194_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_8195_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_8196_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_8197_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_8198_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8199_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8200_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8201_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
              & ( ord_less_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8202_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
              & ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8203_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
              & ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8204_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
              & ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8205_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
              & ( ord_less_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8206_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
              & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8207_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
              & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8208_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A2 ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8209_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A2 ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8210_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8211_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A2 ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8212_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A2 ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8213_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A2 ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8214_even__prod__iff,axiom,
    ! [A2: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A2 ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8215_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > complex,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8216_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8217_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8218_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P6 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P6 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8219_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y2: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
        = Y2 )
     => ( ( ( ord_less_nat @ Xb @ Xa2 )
         => ( Y2 = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa2 )
         => ( Y2
            = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_8220_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F3: nat > nat > nat,A4: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F3 @ A4 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_8221_norm__prod__diff,axiom,
    ! [I6: set_real,Z: real > real,W: real > real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I6 ) @ ( groups1681761925125756287l_real @ W @ I6 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8222_norm__prod__diff,axiom,
    ! [I6: set_int,Z: int > real,W: int > real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I6 ) @ ( groups2316167850115554303t_real @ W @ I6 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8223_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: complex > real,W: complex > real] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I6 ) @ ( groups766887009212190081x_real @ W @ I6 ) ) )
          @ ( groups5808333547571424918x_real
            @ ^ [I3: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8224_norm__prod__diff,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > real,W: product_prod_nat_nat > real] :
      ( ! [I4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups6036352826371341000t_real @ Z @ I6 ) @ ( groups6036352826371341000t_real @ W @ I6 ) ) )
          @ ( groups4567486121110086003t_real
            @ ^ [I3: product_prod_nat_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8225_norm__prod__diff,axiom,
    ! [I6: set_real,Z: real > complex,W: real > complex] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I6 ) @ ( groups713298508707869441omplex @ W @ I6 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8226_norm__prod__diff,axiom,
    ! [I6: set_int,Z: int > complex,W: int > complex] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I6 ) @ ( groups7440179247065528705omplex @ W @ I6 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8227_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: complex > complex,W: complex > complex] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I6 ) @ ( groups3708469109370488835omplex @ W @ I6 ) ) )
          @ ( groups5808333547571424918x_real
            @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8228_norm__prod__diff,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > complex,W: product_prod_nat_nat > complex] :
      ( ! [I4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups8110221916422527690omplex @ Z @ I6 ) @ ( groups8110221916422527690omplex @ W @ I6 ) ) )
          @ ( groups4567486121110086003t_real
            @ ^ [I3: product_prod_nat_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8229_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: nat > real,W: nat > real] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I6 ) @ ( groups129246275422532515t_real @ W @ I6 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8230_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: nat > complex,W: nat > complex] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I4 ) ) @ one_one_real ) )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I4 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I6 ) @ ( groups6464643781859351333omplex @ W @ I6 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_8231_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8232_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8233_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8234_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8235_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8236_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8237_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8238_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8239_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8240_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8241_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_8242_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8243_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8244_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8245_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8246_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8247_pochhammer__Suc__prod,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8248_pochhammer__Suc__prod,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8249_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8250_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8251_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8252_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_8253_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_8254_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_8255_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_8256_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_8257_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_8258_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_8259_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_8260_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_8261_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_8262_pochhammer__prod__rev,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A4: complex,N: nat] :
          ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8263_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N: nat] :
          ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8264_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8265_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8266_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8267_prod_Oin__pairs,axiom,
    ! [G: nat > complex,M: nat,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( times_times_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 ) ) ) ).

% prod.in_pairs
thf(fact_8268_prod_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_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 ) ) ) ).

% prod.in_pairs
thf(fact_8269_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_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 ) ) ) ).

% prod.in_pairs
thf(fact_8270_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_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 ) ) ) ).

% prod.in_pairs
thf(fact_8271_sum__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A4: nat] : ( plus_plus_complex @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_8272_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_8273_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_8274_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_8275_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_8276_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8277_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8278_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8279_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8280_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8281_Maclaurin__zero,axiom,
    ! [X2: real,N2: nat,Diff: nat > complex > real] :
      ( ( X2 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_8282_Maclaurin__zero,axiom,
    ! [X2: real,N2: nat,Diff: nat > real > real] :
      ( ( X2 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_8283_Maclaurin__zero,axiom,
    ! [X2: real,N2: nat,Diff: nat > rat > real] :
      ( ( X2 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8284_Maclaurin__zero,axiom,
    ! [X2: real,N2: nat,Diff: nat > nat > real] :
      ( ( X2 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8285_Maclaurin__zero,axiom,
    ! [X2: real,N2: nat,Diff: nat > int > real] :
      ( ( X2 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_8286_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
              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
              @ ( set_ord_lessThan_nat @ N2 ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8287_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_8288_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_8289_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_8290_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8291_cos__paired,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_real @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      @ ( cos_real @ X2 ) ) ).

% cos_paired
thf(fact_8292_sin__paired,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
      @ ( sin_real @ X2 ) ) ).

% sin_paired
thf(fact_8293_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X2 )
            & ( ( sin_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8294_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [T4: real] :
          ( ( ord_less_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ X2 )
          & ( ( sin_real @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8295_Maclaurin__sin__expansion2,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
      & ( ( sin_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8296_Maclaurin__sin__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( sin_real @ X2 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X2 @ N2 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8297_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8298_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ A2 )
             => ( ( F @ X )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8299_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups708209901874060359at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A2 )
             => ( ( F @ X )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_8300_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8301_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8302_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_8303_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_8304_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_8305_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_8306_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_8307_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_8308_fact__div__fact__le__pow,axiom,
    ! [R: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R ) ) ) @ ( power_power_nat @ N2 @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8309_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
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_8310_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8311_ln__prod,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I6 ) )
          = ( groups8097168146408367636l_real
            @ ^ [X: real] : ( ln_ln_real @ ( F @ X ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_8312_ln__prod,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I6 ) )
          = ( groups8778361861064173332t_real
            @ ^ [X: int] : ( ln_ln_real @ ( F @ X ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_8313_ln__prod,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ I6 )
     => ( ! [I4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups6036352826371341000t_real @ F @ I6 ) )
          = ( groups4567486121110086003t_real
            @ ^ [X: product_prod_nat_nat] : ( ln_ln_real @ ( F @ X ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_8314_ln__prod,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I6 ) )
          = ( groups5808333547571424918x_real
            @ ^ [X: complex] : ( ln_ln_real @ ( F @ X ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_8315_ln__prod,axiom,
    ! [I6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ I6 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I6 ) )
          = ( groups6591440286371151544t_real
            @ ^ [X: nat] : ( ln_ln_real @ ( F @ X ) )
            @ I6 ) ) ) ) ).

% ln_prod
thf(fact_8316_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
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_8317_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_8318_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_8319_tan__double,axiom,
    ! [X2: complex] :
      ( ( ( cos_complex @ X2 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X2 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8320_tan__double,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X2 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8321_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8322_Maclaurin__exp__lt,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
            & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
            & ( ( exp_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8323_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_8324_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8325_or_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ A )
      = A ) ).

% or.idem
thf(fact_8326_or_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ A )
      = A ) ).

% or.idem
thf(fact_8327_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_8328_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_8329_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_8330_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_8331_or_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
      = A ) ).

% or.left_neutral
thf(fact_8332_or_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
      = A ) ).

% or.left_neutral
thf(fact_8333_or_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
      = A ) ).

% or.right_neutral
thf(fact_8334_or_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
      = A ) ).

% or.right_neutral
thf(fact_8335_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_8336_tan__periodic__pi,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ pi ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_pi
thf(fact_8337_exp__less__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% exp_less_cancel_iff
thf(fact_8338_exp__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) ) ) ).

% exp_less_mono
thf(fact_8339_exp__le__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% exp_le_cancel_iff
thf(fact_8340_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_8341_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_8342_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_8343_bit_Odisj__one__left,axiom,
    ! [X2: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_left
thf(fact_8344_bit_Odisj__one__left,axiom,
    ! [X2: int] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_left
thf(fact_8345_bit_Odisj__one__right,axiom,
    ! [X2: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_right
thf(fact_8346_bit_Odisj__one__right,axiom,
    ! [X2: int] :
      ( ( bit_se1409905431419307370or_int @ X2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_right
thf(fact_8347_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_8348_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8349_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_8350_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_8351_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_8352_exp__eq__one__iff,axiom,
    ! [X2: real] :
      ( ( ( exp_real @ X2 )
        = one_one_real )
      = ( X2 = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_8353_or__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% or_nonnegative_int_iff
thf(fact_8354_or__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_8355_or__numerals_I8_J,axiom,
    ! [X2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X2 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X2 ) ) ) ).

% or_numerals(8)
thf(fact_8356_or__numerals_I8_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_numerals(8)
thf(fact_8357_or__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y2 ) ) ) ).

% or_numerals(2)
thf(fact_8358_or__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% or_numerals(2)
thf(fact_8359_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_8360_tan__npi,axiom,
    ! [N2: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_8361_one__less__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% one_less_exp_iff
thf(fact_8362_exp__less__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_8363_one__le__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% one_le_exp_iff
thf(fact_8364_exp__le__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_8365_tan__periodic__n,axiom,
    ! [X2: real,N2: num] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_n
thf(fact_8366_tan__periodic__nat,axiom,
    ! [X2: real,N2: nat] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_nat
thf(fact_8367_exp__ln__iff,axiom,
    ! [X2: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X2 ) )
        = X2 )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% exp_ln_iff
thf(fact_8368_exp__ln,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( exp_real @ ( ln_ln_real @ X2 ) )
        = X2 ) ) ).

% exp_ln
thf(fact_8369_tan__periodic__int,axiom,
    ! [X2: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic_int
thf(fact_8370_or__numerals_I3_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y2 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X2 ) @ ( numeral_numeral_int @ Y2 ) ) ) ) ).

% or_numerals(3)
thf(fact_8371_or__numerals_I3_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X2 ) @ ( numeral_numeral_nat @ Y2 ) ) ) ) ).

% or_numerals(3)
thf(fact_8372_or__numerals_I5_J,axiom,
    ! [X2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X2 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X2 ) ) ) ).

% or_numerals(5)
thf(fact_8373_or__numerals_I5_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_numerals(5)
thf(fact_8374_or__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y2 ) ) ) ).

% or_numerals(1)
thf(fact_8375_or__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% or_numerals(1)
thf(fact_8376_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_8377_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_8378_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_8379_tan__periodic,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic
thf(fact_8380_or__numerals_I7_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X2 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y2 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X2 ) @ ( numeral_numeral_int @ Y2 ) ) ) ) ) ).

% or_numerals(7)
thf(fact_8381_or__numerals_I7_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X2 ) @ ( numeral_numeral_nat @ Y2 ) ) ) ) ) ).

% or_numerals(7)
thf(fact_8382_or__numerals_I6_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y2 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X2 ) @ ( numeral_numeral_int @ Y2 ) ) ) ) ) ).

% or_numerals(6)
thf(fact_8383_or__numerals_I6_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X2 ) @ ( numeral_numeral_nat @ Y2 ) ) ) ) ) ).

% or_numerals(6)
thf(fact_8384_or__numerals_I4_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X2 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y2 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X2 ) @ ( numeral_numeral_int @ Y2 ) ) ) ) ) ).

% or_numerals(4)
thf(fact_8385_or__numerals_I4_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X2 ) @ ( numeral_numeral_nat @ Y2 ) ) ) ) ) ).

% or_numerals(4)
thf(fact_8386_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_8387_norm__exp,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X2 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).

% norm_exp
thf(fact_8388_norm__exp,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X2 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).

% norm_exp
thf(fact_8389_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_8390_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_8391_or_Ocommute,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [A4: int,B4: int] : ( bit_se1409905431419307370or_int @ B4 @ A4 ) ) ) ).

% or.commute
thf(fact_8392_or_Ocommute,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [A4: nat,B4: nat] : ( bit_se1412395901928357646or_nat @ B4 @ A4 ) ) ) ).

% or.commute
thf(fact_8393_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_8394_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_8395_of__int__or__eq,axiom,
    ! [K: int,L2: int] :
      ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
      = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).

% of_int_or_eq
thf(fact_8396_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_8397_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_8398_bit_Odisj__zero__right,axiom,
    ! [X2: int] :
      ( ( bit_se1409905431419307370or_int @ X2 @ zero_zero_int )
      = X2 ) ).

% bit.disj_zero_right
thf(fact_8399_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_8400_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_8401_exp__less__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
     => ( ord_less_real @ X2 @ Y2 ) ) ).

% exp_less_cancel
thf(fact_8402_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_8403_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_8404_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8405_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_8406_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_8407_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_8408_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_8409_exp__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y2 ) ) ).

% exp_total
thf(fact_8410_exp__gt__zero,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_gt_zero
thf(fact_8411_not__exp__less__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_8412_not__exp__le__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_8413_exp__ge__zero,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_ge_zero
thf(fact_8414_binomial__le__pow,axiom,
    ! [R: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R ) @ ( power_power_nat @ N2 @ R ) ) ) ).

% binomial_le_pow
thf(fact_8415_or__greater__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).

% or_greater_eq
thf(fact_8416_OR__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y2 ) ) ) ) ).

% OR_lower
thf(fact_8417_mult__exp__exp,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y2 ) )
      = ( exp_complex @ ( plus_plus_complex @ X2 @ Y2 ) ) ) ).

% mult_exp_exp
thf(fact_8418_mult__exp__exp,axiom,
    ! [X2: real,Y2: real] :
      ( ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
      = ( exp_real @ ( plus_plus_real @ X2 @ Y2 ) ) ) ).

% mult_exp_exp
thf(fact_8419_exp__add__commuting,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( times_times_complex @ X2 @ Y2 )
        = ( times_times_complex @ Y2 @ X2 ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X2 @ Y2 ) )
        = ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y2 ) ) ) ) ).

% exp_add_commuting
thf(fact_8420_exp__add__commuting,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( times_times_real @ X2 @ Y2 )
        = ( times_times_real @ Y2 @ X2 ) )
     => ( ( exp_real @ ( plus_plus_real @ X2 @ Y2 ) )
        = ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) ) ) ) ).

% exp_add_commuting
thf(fact_8421_exp__diff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X2 @ Y2 ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X2 ) @ ( exp_complex @ Y2 ) ) ) ).

% exp_diff
thf(fact_8422_exp__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( exp_real @ ( minus_minus_real @ X2 @ Y2 ) )
      = ( divide_divide_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) ) ) ).

% exp_diff
thf(fact_8423_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_8424_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_8425_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_8426_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_8427_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_8428_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_8429_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_8430_exp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).

% exp_gt_one
thf(fact_8431_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_8432_exp__ge__add__one__self,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).

% exp_ge_add_one_self
thf(fact_8433_exp__minus__inverse,axiom,
    ! [X2: real] :
      ( ( times_times_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_8434_exp__minus__inverse,axiom,
    ! [X2: complex] :
      ( ( times_times_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_8435_exp__of__nat2__mult,axiom,
    ! [X2: complex,N2: nat] :
      ( ( exp_complex @ ( times_times_complex @ X2 @ ( semiri8010041392384452111omplex @ N2 ) ) )
      = ( power_power_complex @ ( exp_complex @ X2 ) @ N2 ) ) ).

% exp_of_nat2_mult
thf(fact_8436_exp__of__nat2__mult,axiom,
    ! [X2: real,N2: nat] :
      ( ( exp_real @ ( times_times_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( power_power_real @ ( exp_real @ X2 ) @ N2 ) ) ).

% exp_of_nat2_mult
thf(fact_8437_exp__of__nat__mult,axiom,
    ! [N2: nat,X2: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ X2 ) )
      = ( power_power_complex @ ( exp_complex @ X2 ) @ N2 ) ) ).

% exp_of_nat_mult
thf(fact_8438_exp__of__nat__mult,axiom,
    ! [N2: nat,X2: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) )
      = ( power_power_real @ ( exp_real @ X2 ) @ N2 ) ) ).

% exp_of_nat_mult
thf(fact_8439_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_8440_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_8441_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_8442_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_8443_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_8444_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_8445_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_8446_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_8447_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_8448_tan__def,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).

% tan_def
thf(fact_8449_tan__def,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).

% tan_def
thf(fact_8450_exp__ge__add__one__self__aux,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_8451_lemma__exp__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y2 )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y2 @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y2 ) ) ) ).

% lemma_exp_total
thf(fact_8452_ln__ge__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ Y2 @ ( ln_ln_real @ X2 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y2 ) @ X2 ) ) ) ).

% ln_ge_iff
thf(fact_8453_ln__x__over__x__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y2 ) @ Y2 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_8454_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_8455_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8456_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8457_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_8458_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_8459_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_8460_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_8461_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_8462_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_8463_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_8464_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_8465_exp__divide__power__eq,axiom,
    ! [N2: nat,X2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X2 @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
        = ( exp_complex @ X2 ) ) ) ).

% exp_divide_power_eq
thf(fact_8466_exp__divide__power__eq,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
        = ( exp_real @ X2 ) ) ) ).

% exp_divide_power_eq
thf(fact_8467_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_8468_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_8469_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_8470_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_8471_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_8472_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_8473_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_8474_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8475_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8476_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8477_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8478_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8479_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8480_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_8481_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_8482_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_8483_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_8484_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_8485_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_8486_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_8487_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_8488_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_8489_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_8490_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_8491_lemma__tan__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ? [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 @ Y2 @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_8492_tan__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).

% tan_gt_zero
thf(fact_8493_OR__upper,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_8494_tan__total,axiom,
    ! [Y2: 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 )
        = Y2 )
      & ! [Y3: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
            & ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y3 )
              = Y2 ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
thf(fact_8495_tan__monotone,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y2 ) @ ( tan_real @ X2 ) ) ) ) ) ).

% tan_monotone
thf(fact_8496_tan__monotone_H,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( 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 ) ) ) )
           => ( ( ord_less_real @ Y2 @ X2 )
              = ( ord_less_real @ ( tan_real @ Y2 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_8497_tan__mono__lt__eq,axiom,
    ! [X2: real,Y2: 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 ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) )
              = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_8498_lemma__tan__total1,axiom,
    ! [Y2: 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 )
        = Y2 ) ) ).

% lemma_tan_total1
thf(fact_8499_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_8500_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_8501_tan__inverse,axiom,
    ! [Y2: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y2 ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 ) ) ) ).

% tan_inverse
thf(fact_8502_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_8503_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_8504_add__tan__eq,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( cos_complex @ X2 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y2 )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y2 ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y2 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8505_add__tan__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8506_exp__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_8507_tan__pos__pi2__le,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_8508_tan__total__pos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ? [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 )
            = Y2 ) ) ) ).

% tan_total_pos
thf(fact_8509_tan__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_8510_tan__mono__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) ) ) ) ).

% tan_mono_le
thf(fact_8511_tan__mono__le__eq,axiom,
    ! [X2: real,Y2: 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 ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) )
              = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_8512_tan__bound__pi2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_8513_arctan__unique,axiom,
    ! [X2: real,Y2: 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 @ Y2 )
            = X2 ) ) ) ) ).

% arctan_unique
thf(fact_8514_arctan__tan,axiom,
    ! [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 ) ) ) )
       => ( ( arctan @ ( tan_real @ X2 ) )
          = X2 ) ) ) ).

% arctan_tan
thf(fact_8515_arctan,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) )
      & ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y2 ) )
        = Y2 ) ) ).

% arctan
thf(fact_8516_lemma__tan__add1,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( cos_complex @ X2 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y2 )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y2 ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y2 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8517_lemma__tan__add1,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8518_tan__diff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( cos_complex @ X2 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y2 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X2 @ Y2 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X2 @ Y2 ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y2 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y2 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8519_tan__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X2 @ Y2 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X2 @ Y2 ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8520_tan__add,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( cos_complex @ X2 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y2 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X2 @ Y2 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X2 @ Y2 ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y2 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X2 ) @ ( tan_complex @ Y2 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8521_tan__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X2 @ Y2 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X2 @ Y2 ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8522_real__exp__bound__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_8523_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X2 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8524_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ X2 @ ( 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 @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8525_tan__total__pi4,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ? [Z4: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
          & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z4 )
            = X2 ) ) ) ).

% tan_total_pi4
thf(fact_8526_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_8527_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_8528_Maclaurin__exp__le,axiom,
    ! [X2: real,N2: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
      & ( ( exp_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8529_exp__lower__Taylor__quadratic,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_8530_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_8531_tan__half,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_8532_tan__half,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_8533_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_8534_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8535_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8536_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8537_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8538_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8539_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8540_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8541_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8542_sin__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X2 )
        = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8543_cos__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X2 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_8544_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_8545_real__sqrt__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ X2 )
        = ( sqrt @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% real_sqrt_eq_iff
thf(fact_8546_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I @ K ) ) ).

% atMost_iff
thf(fact_8547_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
      = ( ord_less_eq_set_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_8548_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I @ K ) ) ).

% atMost_iff
thf(fact_8549_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I @ K ) ) ).

% atMost_iff
thf(fact_8550_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_8551_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_8552_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X2: real] :
      ( ( ( sqrt @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_8553_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8554_real__sqrt__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% real_sqrt_less_iff
thf(fact_8555_real__sqrt__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% real_sqrt_le_iff
thf(fact_8556_real__sqrt__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( sqrt @ X2 )
        = one_one_real )
      = ( X2 = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_8557_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_8558_atMost__subset__iff,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y2 ) )
      = ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% atMost_subset_iff
thf(fact_8559_atMost__subset__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X2 ) @ ( set_ord_atMost_rat @ Y2 ) )
      = ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% atMost_subset_iff
thf(fact_8560_atMost__subset__iff,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X2 ) @ ( set_ord_atMost_num @ Y2 ) )
      = ( ord_less_eq_num @ X2 @ Y2 ) ) ).

% atMost_subset_iff
thf(fact_8561_atMost__subset__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X2 ) @ ( set_ord_atMost_int @ Y2 ) )
      = ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% atMost_subset_iff
thf(fact_8562_atMost__subset__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y2 ) )
      = ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% atMost_subset_iff
thf(fact_8563_real__sqrt__gt__0__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y2 ) )
      = ( ord_less_real @ zero_zero_real @ Y2 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_8564_real__sqrt__lt__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8565_real__sqrt__ge__0__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ).

% real_sqrt_ge_0_iff
thf(fact_8566_real__sqrt__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8567_real__sqrt__lt__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
      = ( ord_less_real @ X2 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_8568_real__sqrt__gt__1__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y2 ) )
      = ( ord_less_real @ one_one_real @ Y2 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_8569_real__sqrt__ge__1__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y2 ) )
      = ( ord_less_eq_real @ one_one_real @ Y2 ) ) ).

% real_sqrt_ge_1_iff
thf(fact_8570_real__sqrt__le__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
      = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_8571_real__sqrt__abs2,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( times_times_real @ X2 @ X2 ) )
      = ( abs_abs_real @ X2 ) ) ).

% real_sqrt_abs2
thf(fact_8572_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_8573_Icc__subset__Iic__iff,axiom,
    ! [L2: set_nat,H2: set_nat,H3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L2 @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
      = ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
        | ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8574_Icc__subset__Iic__iff,axiom,
    ! [L2: rat,H2: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L2 @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L2 @ H2 )
        | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8575_Icc__subset__Iic__iff,axiom,
    ! [L2: num,H2: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L2 @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L2 @ H2 )
        | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8576_Icc__subset__Iic__iff,axiom,
    ! [L2: nat,H2: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L2 @ H2 )
        | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8577_Icc__subset__Iic__iff,axiom,
    ! [L2: int,H2: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L2 @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L2 @ H2 )
        | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8578_Icc__subset__Iic__iff,axiom,
    ! [L2: real,H2: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L2 @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L2 @ H2 )
        | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8579_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_8580_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8581_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8582_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8583_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8584_prod_OatMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8585_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8586_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8587_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8588_or__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(4)
thf(fact_8589_or__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% or_nat_numerals(2)
thf(fact_8590_real__sqrt__abs,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X2 ) ) ).

% real_sqrt_abs
thf(fact_8591_or__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% or_nat_numerals(1)
thf(fact_8592_or__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(3)
thf(fact_8593_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_8594_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_8595_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_8596_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_8597_real__sqrt__pow2__iff,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% real_sqrt_pow2_iff
thf(fact_8598_real__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 ) ) ).

% real_sqrt_pow2
thf(fact_8599_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X2: real,Y2: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( 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 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( 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_8600_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_8601_real__sqrt__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( sqrt @ ( times_times_real @ X2 @ Y2 ) )
      = ( times_times_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_mult
thf(fact_8602_real__sqrt__power,axiom,
    ! [X2: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X2 @ K ) )
      = ( power_power_real @ ( sqrt @ X2 ) @ K ) ) ).

% real_sqrt_power
thf(fact_8603_real__sqrt__minus,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X2 ) )
      = ( uminus_uminus_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_minus
thf(fact_8604_real__sqrt__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_less_mono
thf(fact_8605_real__sqrt__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_le_mono
thf(fact_8606_real__sqrt__divide,axiom,
    ! [X2: real,Y2: real] :
      ( ( sqrt @ ( divide_divide_real @ X2 @ Y2 ) )
      = ( divide_divide_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_divide
thf(fact_8607_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_8608_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8609_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_eq_set_nat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8610_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8611_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8612_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8613_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8614_real__sqrt__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8615_real__sqrt__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8616_real__sqrt__eq__zero__cancel,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( sqrt @ X2 )
          = zero_zero_real )
       => ( X2 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8617_real__sqrt__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_one
thf(fact_8618_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_8619_not__Iic__le__Icc,axiom,
    ! [H2: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_8620_not__Iic__le__Icc,axiom,
    ! [H2: real,L3: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_8621_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_8622_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_8623_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_8624_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_8625_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_8626_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_8627_real__div__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
        = ( sqrt @ X2 ) ) ) ).

% real_div_sqrt
thf(fact_8628_sqrt__add__le__add__sqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8629_le__real__sqrt__sumsq,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8630_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8631_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8632_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8633_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8634_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_8635_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_8636_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_8637_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8638_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_8639_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8640_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8641_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8642_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8643_sum__telescope,axiom,
    ! [F: nat > complex,I: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( minus_minus_complex @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8644_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8645_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8646_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N2: nat,D: nat > complex] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8647_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N2: nat,D: nat > real] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8648_bounded__imp__summable,axiom,
    ! [A: nat > int,B2: int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N4 ) ) @ B2 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8649_bounded__imp__summable,axiom,
    ! [A: nat > nat,B2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N4 ) ) @ B2 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8650_bounded__imp__summable,axiom,
    ! [A: nat > real,B2: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N4 ) ) @ B2 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8651_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8652_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8653_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8654_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8655_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_8656_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_8657_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8658_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_8659_sum__choose__lower,axiom,
    ! [R: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_8660_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_8661_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_8662_real__less__rsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
     => ( ord_less_real @ X2 @ ( sqrt @ Y2 ) ) ) ).

% real_less_rsqrt
thf(fact_8663_real__le__rsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
     => ( ord_less_eq_real @ X2 @ ( sqrt @ Y2 ) ) ) ).

% real_le_rsqrt
thf(fact_8664_sqrt__le__D,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y2 )
     => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_8665_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_8666_polyfun__eq__0,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_complex ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_8667_polyfun__eq__0,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_real ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_8668_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N2: nat,K: nat] :
      ( ! [W2: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8669_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N2: nat,K: nat] :
      ( ! [W2: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8670_or__not__num__neg_Oelims,axiom,
    ! [X2: num,Xa2: num,Y2: num] :
      ( ( ( bit_or_not_num_neg @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( X2 = one )
         => ( ( Xa2 = one )
           => ( Y2 != one ) ) )
       => ( ( ( X2 = one )
           => ! [M5: num] :
                ( ( Xa2
                  = ( bit0 @ M5 ) )
               => ( Y2
                 != ( bit1 @ M5 ) ) ) )
         => ( ( ( X2 = one )
             => ! [M5: num] :
                  ( ( Xa2
                    = ( bit1 @ M5 ) )
                 => ( Y2
                   != ( bit1 @ M5 ) ) ) )
           => ( ( ? [N4: num] :
                    ( X2
                    = ( bit0 @ N4 ) )
               => ( ( Xa2 = one )
                 => ( Y2
                   != ( bit0 @ one ) ) ) )
             => ( ! [N4: num] :
                    ( ( X2
                      = ( bit0 @ N4 ) )
                   => ! [M5: num] :
                        ( ( Xa2
                          = ( bit0 @ M5 ) )
                       => ( Y2
                         != ( bitM @ ( bit_or_not_num_neg @ N4 @ M5 ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X2
                        = ( bit0 @ N4 ) )
                     => ! [M5: num] :
                          ( ( Xa2
                            = ( bit1 @ M5 ) )
                         => ( Y2
                           != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M5 ) ) ) ) )
                 => ( ( ? [N4: num] :
                          ( X2
                          = ( bit1 @ N4 ) )
                     => ( ( Xa2 = one )
                       => ( Y2 != one ) ) )
                   => ( ! [N4: num] :
                          ( ( X2
                            = ( bit1 @ N4 ) )
                         => ! [M5: num] :
                              ( ( Xa2
                                = ( bit0 @ M5 ) )
                             => ( Y2
                               != ( bitM @ ( bit_or_not_num_neg @ N4 @ M5 ) ) ) ) )
                     => ~ ! [N4: num] :
                            ( ( X2
                              = ( bit1 @ N4 ) )
                           => ! [M5: num] :
                                ( ( Xa2
                                  = ( bit1 @ M5 ) )
                               => ( Y2
                                 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_8671_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8672_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8673_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8674_sum_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_8675_sum__up__index__split,axiom,
    ! [F: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8676_sum__up__index__split,axiom,
    ! [F: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8677_sum__up__index__split,axiom,
    ! [F: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8678_sum__up__index__split,axiom,
    ! [F: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8679_prod_OatMost__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8680_prod_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8681_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8682_prod_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_8683_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_8684_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > real,N2: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_8685_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8686_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8687_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_8688_vandermonde,axiom,
    ! [M: nat,N2: nat,R: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R ) ) ).

% vandermonde
thf(fact_8689_real__sqrt__unique,axiom,
    ! [Y2: real,X2: real] :
      ( ( ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( sqrt @ X2 )
          = Y2 ) ) ) ).

% real_sqrt_unique
thf(fact_8690_real__le__lsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).

% real_le_lsqrt
thf(fact_8691_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_8692_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y2 )
     => ( X2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_8693_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X2 )
     => ( Y2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_8694_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_8695_real__sqrt__sum__squares__ge2,axiom,
    ! [Y2: real,X2: real] : ( ord_less_eq_real @ Y2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_8696_real__sqrt__sum__squares__ge1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_8697_sqrt__ge__absD,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y2 ) )
     => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) ).

% sqrt_ge_absD
thf(fact_8698_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_8699_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_8700_sum__gp__basic,axiom,
    ! [X2: rat,N2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8701_sum__gp__basic,axiom,
    ! [X2: complex,N2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8702_sum__gp__basic,axiom,
    ! [X2: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8703_sum__gp__basic,axiom,
    ! [X2: real,N2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8704_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_complex ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
            & ( ( C @ I3 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_8705_polyfun__finite__roots,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_real ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
            & ( ( C @ I3 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_8706_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_8707_polyfun__roots__finite,axiom,
    ! [C: nat > real,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z5: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_8708_polyfun__linear__factor__root,axiom,
    ! [C: nat > rat,A: rat,N2: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat )
     => ~ ! [B3: nat > rat] :
            ~ ! [Z3: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I3: nat] : ( times_times_rat @ ( B3 @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8709_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N2: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex )
     => ~ ! [B3: nat > complex] :
            ~ ! [Z3: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I3: nat] : ( times_times_complex @ ( B3 @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8710_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N2: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int )
     => ~ ! [B3: nat > int] :
            ~ ! [Z3: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_int @ ( minus_minus_int @ Z3 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I3: nat] : ( times_times_int @ ( B3 @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8711_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N2: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real )
     => ~ ! [B3: nat > real] :
            ~ ! [Z3: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_real @ ( minus_minus_real @ Z3 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( B3 @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8712_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N2: nat,A: rat] :
    ? [B3: nat > rat] :
    ! [Z3: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( B3 @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8713_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N2: nat,A: complex] :
    ? [B3: nat > complex] :
    ! [Z3: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( B3 @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8714_polyfun__linear__factor,axiom,
    ! [C: nat > int,N2: nat,A: int] :
    ? [B3: nat > int] :
    ! [Z3: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z3 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( B3 @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8715_polyfun__linear__factor,axiom,
    ! [C: nat > real,N2: nat,A: real] :
    ? [B3: nat > real] :
    ! [Z3: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z3 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( B3 @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8716_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X2: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8717_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X2: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8718_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8719_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_8720_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > real,N2: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_8721_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_8722_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_8723_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_8724_summable__Cauchy__product,axiom,
    ! [A: nat > complex,B: nat > complex] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
       => ( summable_complex
          @ ^ [K3: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_8725_summable__Cauchy__product,axiom,
    ! [A: nat > real,B: nat > real] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
       => ( summable_real
          @ ^ [K3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_8726_Cauchy__product,axiom,
    ! [A: nat > complex,B: nat > complex] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
       => ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) )
          = ( suminf_complex
            @ ^ [K3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
                @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_8727_Cauchy__product,axiom,
    ! [A: nat > real,B: nat > real] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
       => ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) )
          = ( suminf_real
            @ ^ [K3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
                @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_8728_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_8729_real__less__lsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_real @ X2 @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).

% real_less_lsqrt
thf(fact_8730_sqrt__sum__squares__le__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y2 ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8731_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_8732_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_8733_real__sqrt__ge__abs1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8734_real__sqrt__ge__abs2,axiom,
    ! [Y2: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8735_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y2 ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8736_ln__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( sqrt @ X2 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8737_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_8738_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_8739_complex__norm,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X2 @ Y2 ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_8740_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( 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_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8741_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( 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_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8742_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( 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_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8743_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( 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_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8744_polynomial__product,axiom,
    ! [M: nat,A: nat > rat,N2: nat,B: nat > rat,X2: rat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R4: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R4 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_rat @ X2 @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8745_polynomial__product,axiom,
    ! [M: nat,A: nat > complex,N2: nat,B: nat > complex,X2: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R4: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R4 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_complex @ X2 @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8746_polynomial__product,axiom,
    ! [M: nat,A: nat > int,N2: nat,B: nat > int,X2: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R4: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R4 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_int @ X2 @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8747_polynomial__product,axiom,
    ! [M: nat,A: nat > real,N2: nat,B: nat > real,X2: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R4: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R4 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_real @ X2 @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8748_prod_Oin__pairs__0,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8749_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8750_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8751_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_8752_polyfun__eq__const,axiom,
    ! [C: nat > complex,N2: nat,K: complex] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_8753_polyfun__eq__const,axiom,
    ! [C: nat > real,N2: nat,K: real] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_8754_binomial__ring,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8755_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8756_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8757_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8758_binomial__ring,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8759_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K3 ) ) @ ( comm_s2602460028002588243omplex @ A @ K3 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8760_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8761_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8762_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8763_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X2: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = 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 @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R4: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R4 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_nat @ X2 @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8764_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N2 @ K3 ) @ ( 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_8765_Cauchy__product__sums,axiom,
    ! [A: nat > complex,B: nat > complex] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
       => ( sums_complex
          @ ^ [K3: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) )
          @ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_8766_Cauchy__product__sums,axiom,
    ! [A: nat > real,B: nat > real] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
       => ( sums_real
          @ ^ [K3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) )
          @ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_8767_arsinh__real__aux,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8768_real__sqrt__power__even,axiom,
    ! [N2: nat,X2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( sqrt @ X2 ) @ N2 )
          = ( power_power_real @ X2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8769_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X2: real,Y2: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( 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_8770_arith__geo__mean__sqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y2 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8771_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_8772_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_8773_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_8774_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8775_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8776_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8777_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8778_sum_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8779_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8780_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8781_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8782_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8783_prod_Ozero__middle,axiom,
    ! [P6: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P6 )
     => ( ( ord_less_eq_nat @ K @ P6 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P6 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8784_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_8785_cos__x__y__le__one,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8786_real__sqrt__sum__squares__less,axiom,
    ! [X2: real,U: real,Y2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8787_arcosh__real__def,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( arcosh_real @ X2 )
        = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8788_cos__arctan,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( arctan @ X2 ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_8789_sin__arctan,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( arctan @ X2 ) )
      = ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_8790_root__polyfun,axiom,
    ! [N2: nat,Z: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_int @ Z @ N2 )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N2 ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_8791_root__polyfun,axiom,
    ! [N2: nat,Z: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_complex @ Z @ N2 )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N2 ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_8792_root__polyfun,axiom,
    ! [N2: nat,Z: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_8256067586552552935nteger @ Z @ N2 )
          = A )
        = ( ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I3 = N2 ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_8793_root__polyfun,axiom,
    ! [N2: nat,Z: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_rat @ Z @ N2 )
          = A )
        = ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I3 = N2 ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_8794_root__polyfun,axiom,
    ! [N2: nat,Z: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_real @ Z @ N2 )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N2 ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_8795_sum__gp0,axiom,
    ! [X2: complex,N2: nat] :
      ( ( ( X2 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X2 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).

% sum_gp0
thf(fact_8796_sum__gp0,axiom,
    ! [X2: rat,N2: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp0
thf(fact_8797_sum__gp0,axiom,
    ! [X2: real,N2: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp0
thf(fact_8798_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_8799_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_8800_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_8801_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_8802_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_8803_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > complex,X2: complex,Y2: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y2 )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y2 @ K3 ) ) @ ( power_power_complex @ X2 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8804_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > int,X2: int,Y2: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X2 @ Y2 )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y2 @ K3 ) ) @ ( power_power_int @ X2 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8805_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > real,X2: real,Y2: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X2 @ Y2 )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y2 @ K3 ) ) @ ( power_power_real @ X2 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8806_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_8807_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_8808_sqrt__sum__squares__half__less,axiom,
    ! [X2: real,U: real,Y2: real] :
      ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8809_sin__cos__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
     => ( ( sin_real @ X2 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8810_arctan__half,axiom,
    ( arctan
    = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8811_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_8812_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_8813_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_8814_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_8815_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_8816_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > complex,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z3: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z3 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8817_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z3: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z3 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z3 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8818_polyfun__diff,axiom,
    ! [N2: nat,A: nat > complex,X2: complex,Y2: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y2 )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_complex @ X2 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8819_polyfun__diff,axiom,
    ! [N2: nat,A: nat > int,X2: int,Y2: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X2 @ Y2 )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_int @ X2 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8820_polyfun__diff,axiom,
    ! [N2: nat,A: nat > real,X2: real,Y2: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X2 @ Y2 )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_real @ X2 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_8821_sin__x__sin__y,axiom,
    ! [X2: real,Y2: real] :
      ( sums_real
      @ ^ [P4: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) ) @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ P4 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P4 ) )
      @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) ) ) ).

% sin_x_sin_y
thf(fact_8822_sin__x__sin__y,axiom,
    ! [X2: complex,Y2: complex] :
      ( sums_complex
      @ ^ [P4: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) ) @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ P4 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P4 ) )
      @ ( times_times_complex @ ( sin_complex @ X2 ) @ ( sin_complex @ Y2 ) ) ) ).

% sin_x_sin_y
thf(fact_8823_sums__cos__x__plus__y,axiom,
    ! [X2: real,Y2: real] :
      ( sums_real
      @ ^ [P4: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ P4 @ N ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P4 ) )
      @ ( cos_real @ ( plus_plus_real @ X2 @ Y2 ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8824_sums__cos__x__plus__y,axiom,
    ! [X2: complex,Y2: complex] :
      ( sums_complex
      @ ^ [P4: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ P4 @ N ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P4 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X2 @ Y2 ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8825_cos__x__cos__y,axiom,
    ! [X2: real,Y2: real] :
      ( sums_real
      @ ^ [P4: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ P4 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P4 ) )
      @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ).

% cos_x_cos_y
thf(fact_8826_cos__x__cos__y,axiom,
    ! [X2: complex,Y2: complex] :
      ( sums_complex
      @ ^ [P4: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ P4 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P4 ) )
      @ ( times_times_complex @ ( cos_complex @ X2 ) @ ( cos_complex @ Y2 ) ) ) ).

% cos_x_cos_y
thf(fact_8827_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8828_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8829_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8830_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8831_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_8832_mult__scaleR__left,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ Y2 )
      = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X2 @ Y2 ) ) ) ).

% mult_scaleR_left
thf(fact_8833_mult__scaleR__left,axiom,
    ! [A: real,X2: complex,Y2: complex] :
      ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X2 ) @ Y2 )
      = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X2 @ Y2 ) ) ) ).

% mult_scaleR_left
thf(fact_8834_mult__scaleR__right,axiom,
    ! [X2: real,A: real,Y2: real] :
      ( ( times_times_real @ X2 @ ( real_V1485227260804924795R_real @ A @ Y2 ) )
      = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X2 @ Y2 ) ) ) ).

% mult_scaleR_right
thf(fact_8835_mult__scaleR__right,axiom,
    ! [X2: complex,A: real,Y2: complex] :
      ( ( times_times_complex @ X2 @ ( real_V2046097035970521341omplex @ A @ Y2 ) )
      = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X2 @ Y2 ) ) ) ).

% mult_scaleR_right
thf(fact_8836_scaleR__one,axiom,
    ! [X2: real] :
      ( ( real_V1485227260804924795R_real @ one_one_real @ X2 )
      = X2 ) ).

% scaleR_one
thf(fact_8837_scaleR__one,axiom,
    ! [X2: complex] :
      ( ( real_V2046097035970521341omplex @ one_one_real @ X2 )
      = X2 ) ).

% scaleR_one
thf(fact_8838_scaleR__scaleR,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X2 ) )
      = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).

% scaleR_scaleR
thf(fact_8839_scaleR__scaleR,axiom,
    ! [A: real,B: real,X2: complex] :
      ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X2 ) )
      = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X2 ) ) ).

% scaleR_scaleR
thf(fact_8840_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
        = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
      = ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_8841_scaleR__eq__iff,axiom,
    ! [B: complex,U: real,A: complex] :
      ( ( ( plus_plus_complex @ B @ ( real_V2046097035970521341omplex @ U @ A ) )
        = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B ) ) )
      = ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_8842_scaleR__power,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( power_power_real @ ( real_V1485227260804924795R_real @ X2 @ Y2 ) @ N2 )
      = ( real_V1485227260804924795R_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) ) ) ).

% scaleR_power
thf(fact_8843_scaleR__power,axiom,
    ! [X2: real,Y2: complex,N2: nat] :
      ( ( power_power_complex @ ( real_V2046097035970521341omplex @ X2 @ Y2 ) @ N2 )
      = ( real_V2046097035970521341omplex @ ( power_power_real @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ N2 ) ) ) ).

% scaleR_power
thf(fact_8844_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_8845_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_8846_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_8847_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_8848_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_8849_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_8850_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_8851_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_8852_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_8853_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_8854_scaleR__minus1__left,axiom,
    ! [X2: real] :
      ( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
      = ( uminus_uminus_real @ X2 ) ) ).

% scaleR_minus1_left
thf(fact_8855_scaleR__minus1__left,axiom,
    ! [X2: complex] :
      ( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X2 )
      = ( uminus1482373934393186551omplex @ X2 ) ) ).

% scaleR_minus1_left
thf(fact_8856_scaleR__collapse,axiom,
    ! [U: real,A: real] :
      ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_8857_scaleR__collapse,axiom,
    ! [U: real,A: complex] :
      ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_8858_norm__scaleR,axiom,
    ! [A: real,X2: real] :
      ( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X2 ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).

% norm_scaleR
thf(fact_8859_norm__scaleR,axiom,
    ! [A: real,X2: complex] :
      ( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X2 ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).

% norm_scaleR
thf(fact_8860_scaleR__times,axiom,
    ! [U: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).

% scaleR_times
thf(fact_8861_scaleR__times,axiom,
    ! [U: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).

% scaleR_times
thf(fact_8862_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% inverse_scaleR_times
thf(fact_8863_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% inverse_scaleR_times
thf(fact_8864_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% fraction_scaleR_times
thf(fact_8865_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% fraction_scaleR_times
thf(fact_8866_scaleR__half__double,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_8867_scaleR__half__double,axiom,
    ! [A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_8868_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8869_scaleR__right__distrib,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ ( real_V1485227260804924795R_real @ A @ Y2 ) ) ) ).

% scaleR_right_distrib
thf(fact_8870_scaleR__right__distrib,axiom,
    ! [A: real,X2: complex,Y2: complex] :
      ( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X2 @ Y2 ) )
      = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X2 ) @ ( real_V2046097035970521341omplex @ A @ Y2 ) ) ) ).

% scaleR_right_distrib
thf(fact_8871_scaleR__left_Oadd,axiom,
    ! [X2: real,Y2: real,Xa2: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X2 @ Y2 ) @ Xa2 )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X2 @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y2 @ Xa2 ) ) ) ).

% scaleR_left.add
thf(fact_8872_scaleR__left_Oadd,axiom,
    ! [X2: real,Y2: real,Xa2: complex] :
      ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X2 @ Y2 ) @ Xa2 )
      = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X2 @ Xa2 ) @ ( real_V2046097035970521341omplex @ Y2 @ Xa2 ) ) ) ).

% scaleR_left.add
thf(fact_8873_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X2 )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ ( real_V1485227260804924795R_real @ B @ X2 ) ) ) ).

% scaleR_left_distrib
thf(fact_8874_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X2: complex] :
      ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X2 )
      = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X2 ) @ ( real_V2046097035970521341omplex @ B @ X2 ) ) ) ).

% scaleR_left_distrib
thf(fact_8875_complex__scaleR,axiom,
    ! [R: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).

% complex_scaleR
thf(fact_8876_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).

% scaleR_right_mono_neg
thf(fact_8877_scaleR__right__mono,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ ( real_V1485227260804924795R_real @ B @ X2 ) ) ) ) ).

% scaleR_right_mono
thf(fact_8878_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_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 ) ) ) ) ).

% scaleR_le_cancel_left
thf(fact_8879_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% scaleR_le_cancel_left_neg
thf(fact_8880_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% scaleR_le_cancel_left_pos
thf(fact_8881_scaleR__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 @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).

% scaleR_left_mono_neg
thf(fact_8882_scaleR__left__mono,axiom,
    ! [X2: real,Y2: real,A: real] :
      ( ( ord_less_eq_real @ X2 @ Y2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ ( real_V1485227260804924795R_real @ A @ Y2 ) ) ) ) ).

% scaleR_left_mono
thf(fact_8883_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8884_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8885_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8886_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X2: real] :
      ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
        = X2 )
      = ( ( ( V = zero_zero_real )
         => ( X2 = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ U @ A )
            = ( real_V1485227260804924795R_real @ V @ X2 ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_8887_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,X2: complex] :
      ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
        = X2 )
      = ( ( ( V = zero_zero_real )
         => ( X2 = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V2046097035970521341omplex @ U @ A )
            = ( real_V2046097035970521341omplex @ V @ X2 ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_8888_eq__vector__fraction__iff,axiom,
    ! [X2: real,U: real,V: real,A: real] :
      ( ( X2
        = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X2 = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ V @ X2 )
            = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_8889_eq__vector__fraction__iff,axiom,
    ! [X2: complex,U: real,V: real,A: complex] :
      ( ( X2
        = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X2 = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V2046097035970521341omplex @ V @ X2 )
            = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_8890_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_8891_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_8892_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N2 ) @ K )
        = ( gbinomial_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_8893_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_8894_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K )
        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_8895_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
thf(fact_8896_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
thf(fact_8897_scaleR__mono,axiom,
    ! [A: real,B: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ ( real_V1485227260804924795R_real @ B @ Y2 ) ) ) ) ) ) ).

% scaleR_mono
thf(fact_8898_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).

% scaleR_mono'
thf(fact_8899_split__scaleR__neg__le,axiom,
    ! [A: real,X2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ X2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ X2 ) ) )
     => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ zero_zero_real ) ) ).

% split_scaleR_neg_le
thf(fact_8900_split__scaleR__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 @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).

% split_scaleR_pos_le
thf(fact_8901_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X2 ) ) ) ) ).

% scaleR_nonneg_nonneg
thf(fact_8902_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ zero_zero_real ) ) ) ).

% scaleR_nonneg_nonpos
thf(fact_8903_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ zero_zero_real ) ) ) ).

% scaleR_nonpos_nonneg
thf(fact_8904_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).

% scaleR_nonpos_nonpos
thf(fact_8905_scaleR__left__le__one__le,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X2 ) @ X2 ) ) ) ).

% scaleR_left_le_one_le
thf(fact_8906_scaleR__2,axiom,
    ! [X2: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 )
      = ( plus_plus_real @ X2 @ X2 ) ) ).

% scaleR_2
thf(fact_8907_scaleR__2,axiom,
    ! [X2: complex] :
      ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 )
      = ( plus_plus_complex @ X2 @ X2 ) ) ).

% scaleR_2
thf(fact_8908_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8909_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8910_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8911_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8912_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8913_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8914_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8915_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8916_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8917_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8918_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8919_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8920_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8921_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8922_sin__converges,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ X2 @ N ) )
      @ ( sin_real @ X2 ) ) ).

% sin_converges
thf(fact_8923_sin__converges,axiom,
    ! [X2: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ X2 @ N ) )
      @ ( sin_complex @ X2 ) ) ).

% sin_converges
thf(fact_8924_sin__def,axiom,
    ( sin_real
    = ( ^ [X: real] :
          ( suminf_real
          @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% sin_def
thf(fact_8925_sin__def,axiom,
    ( sin_complex
    = ( ^ [X: complex] :
          ( suminf_complex
          @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% sin_def
thf(fact_8926_cos__converges,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ X2 @ N ) )
      @ ( cos_real @ X2 ) ) ).

% cos_converges
thf(fact_8927_cos__converges,axiom,
    ! [X2: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ X2 @ N ) )
      @ ( cos_complex @ X2 ) ) ).

% cos_converges
thf(fact_8928_cos__def,axiom,
    ( cos_real
    = ( ^ [X: real] :
          ( suminf_real
          @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% cos_def
thf(fact_8929_cos__def,axiom,
    ( cos_complex
    = ( ^ [X: complex] :
          ( suminf_complex
          @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% cos_def
thf(fact_8930_summable__norm__sin,axiom,
    ! [X2: real] :
      ( summable_real
      @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ X2 @ N ) ) ) ) ).

% summable_norm_sin
thf(fact_8931_summable__norm__sin,axiom,
    ! [X2: complex] :
      ( summable_real
      @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ X2 @ N ) ) ) ) ).

% summable_norm_sin
thf(fact_8932_summable__norm__cos,axiom,
    ! [X2: real] :
      ( summable_real
      @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ X2 @ N ) ) ) ) ).

% summable_norm_cos
thf(fact_8933_summable__norm__cos,axiom,
    ! [X2: complex] :
      ( summable_real
      @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ X2 @ N ) ) ) ) ).

% summable_norm_cos
thf(fact_8934_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8935_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8936_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8937_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8938_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8939_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8940_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8941_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8942_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8943_gbinomial__parallel__sum,axiom,
    ! [A: complex,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8944_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8945_gbinomial__parallel__sum,axiom,
    ! [A: real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8946_sin__minus__converges,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ ( uminus_uminus_real @ X2 ) @ N ) ) )
      @ ( sin_real @ X2 ) ) ).

% sin_minus_converges
thf(fact_8947_sin__minus__converges,axiom,
    ! [X2: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ N ) ) )
      @ ( sin_complex @ X2 ) ) ).

% sin_minus_converges
thf(fact_8948_cos__minus__converges,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ ( uminus_uminus_real @ X2 ) @ N ) )
      @ ( cos_real @ X2 ) ) ).

% cos_minus_converges
thf(fact_8949_cos__minus__converges,axiom,
    ! [X2: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ N ) )
      @ ( cos_complex @ X2 ) ) ).

% cos_minus_converges
thf(fact_8950_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8951_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8952_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8953_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8954_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8955_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8956_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8957_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8958_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8959_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ K ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_8960_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ K ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_8961_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_8962_gbinomial__minus,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8963_gbinomial__minus,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8964_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8965_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8966_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8967_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8968_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8969_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8970_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8971_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8972_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8973_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8974_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8975_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8976_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8977_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X2: complex,Y2: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X2 @ K3 ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X2 @ Y2 ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8978_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X2: rat,Y2: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X2 @ K3 ) ) @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8979_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X2: real,Y2: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X2 @ K3 ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X2 ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8980_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8981_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8982_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8983_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8984_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8985_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8986_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8987_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8988_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8989_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8990_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8991_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8992_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8993_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8994_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X2: complex,Y2: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X2 @ K3 ) ) @ ( power_power_complex @ Y2 @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X2 @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X2 @ Y2 ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8995_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X2: rat,Y2: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X2 @ K3 ) ) @ ( power_power_rat @ Y2 @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X2 @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8996_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X2: real,Y2: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X2 @ K3 ) ) @ ( power_power_real @ Y2 @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X2 @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8997_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8998_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8999_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_9000_gchoose__row__sum__weighted,axiom,
    ! [R: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_9001_gchoose__row__sum__weighted,axiom,
    ! [R: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_9002_gchoose__row__sum__weighted,axiom,
    ! [R: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_9003_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_9004_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_9005_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_9006_cos__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X2 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_9007_sin__arccos__abs,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y2 ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_9008_sin__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X2 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_9009_exp__first__two__terms,axiom,
    ( exp_complex
    = ( ^ [X: complex] :
          ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X )
          @ ( suminf_complex
            @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_9010_Maclaurin__sin__bound,axiom,
    ! [X2: real,N2: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X2 )
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X2 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N2 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9011_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9012_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_9013_cos__arccos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y2 ) )
          = Y2 ) ) ) ).

% cos_arccos
thf(fact_9014_sin__arcsin,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y2 ) )
          = Y2 ) ) ) ).

% sin_arcsin
thf(fact_9015_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9016_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_9017_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_9018_real__sqrt__inverse,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X2 ) )
      = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_inverse
thf(fact_9019_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).

% divide_real_def
thf(fact_9020_arccos__le__arccos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_9021_arccos__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
          = ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% arccos_le_mono
thf(fact_9022_arccos__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real ) )
     => ( ( ( arccos @ X2 )
          = ( arccos @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% arccos_eq_iff
thf(fact_9023_arcsin__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
          = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).

% arcsin_minus
thf(fact_9024_arcsin__le__arcsin,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_9025_arcsin__le__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
          = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).

% arcsin_le_mono
thf(fact_9026_arcsin__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ( arcsin @ X2 )
            = ( arcsin @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% arcsin_eq_iff
thf(fact_9027_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9028_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N4: nat] :
            ( ( N4 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_9029_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N: nat] :
            ( ( N != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_9030_sqrt__divide__self__eq,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
        = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9031_ln__inverse,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X2 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_inverse
thf(fact_9032_arccos__lbound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) ) ) ) ).

% arccos_lbound
thf(fact_9033_arccos__less__arccos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_9034_arccos__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
          = ( ord_less_real @ Y2 @ X2 ) ) ) ) ).

% arccos_less_mono
thf(fact_9035_arccos__ubound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_9036_arccos__cos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( arccos @ ( cos_real @ X2 ) )
          = X2 ) ) ) ).

% arccos_cos
thf(fact_9037_arcsin__less__arcsin,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_9038_arcsin__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
          = ( ord_less_real @ X2 @ Y2 ) ) ) ) ).

% arcsin_less_mono
thf(fact_9039_cos__arccos__abs,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y2 ) )
        = Y2 ) ) ).

% cos_arccos_abs
thf(fact_9040_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_9041_arccos__lt__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y2 ) )
          & ( ord_less_real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_9042_arccos__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) )
          & ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_9043_sin__arccos__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X2 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_9044_arccos__cos2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
       => ( ( arccos @ ( cos_real @ X2 ) )
          = ( uminus_uminus_real @ X2 ) ) ) ) ).

% arccos_cos2
thf(fact_9045_arccos__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_minus
thf(fact_9046_cos__arcsin__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X2 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_9047_exp__plus__inverse__exp,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9048_plus__inverse__ge__2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9049_real__inv__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X2 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9050_arccos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) )
          & ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi )
          & ( ( cos_real @ ( arccos @ Y2 ) )
            = Y2 ) ) ) ) ).

% arccos
thf(fact_9051_arccos__minus__abs,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).

% arccos_minus_abs
thf(fact_9052_tan__cot,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
      = ( inverse_inverse_real @ ( tan_real @ X2 ) ) ) ).

% tan_cot
thf(fact_9053_real__le__x__sinh,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9054_real__le__abs__sinh,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9055_arccos__le__pi2,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_9056_arcsin__lt__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_9057_arcsin__lbound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) ) ) ) ).

% arcsin_lbound
thf(fact_9058_arcsin__ubound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_9059_arcsin__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_9060_arcsin__sin,axiom,
    ! [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 ) ) ) )
       => ( ( arcsin @ ( sin_real @ X2 ) )
          = X2 ) ) ) ).

% arcsin_sin
thf(fact_9061_arcsin,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y2 ) )
            = Y2 ) ) ) ) ).

% arcsin
thf(fact_9062_arcsin__pi,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y2 ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y2 ) )
            = Y2 ) ) ) ) ).

% arcsin_pi
thf(fact_9063_arcsin__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y2 )
              = ( ord_less_eq_real @ X2 @ ( sin_real @ Y2 ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_9064_le__arcsin__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y2 @ ( arcsin @ X2 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y2 ) @ X2 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_9065_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_9066_cot__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9067_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_9068_log__base__10__eq1,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
        = ( 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 @ X2 ) ) ) ) ).

% log_base_10_eq1
thf(fact_9069_sinh__real__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% sinh_real_less_iff
thf(fact_9070_sinh__real__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y2 ) )
      = ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% sinh_real_le_iff
thf(fact_9071_sinh__real__pos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% sinh_real_pos_iff
thf(fact_9072_sinh__real__neg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sinh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9073_sinh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% sinh_real_nonneg_iff
thf(fact_9074_sinh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9075_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_9076_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_9077_zero__less__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X2 ) )
          = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_9078_log__less__zero__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A @ X2 ) @ zero_zero_real )
          = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_9079_one__less__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X2 ) )
          = ( ord_less_real @ A @ X2 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_9080_log__less__one__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A @ X2 ) @ one_one_real )
          = ( ord_less_real @ X2 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_9081_log__less__cancel__iff,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) )
            = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_9082_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_9083_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_9084_log__le__cancel__iff,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) )
            = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_9085_log__le__one__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ one_one_real )
          = ( ord_less_eq_real @ X2 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_9086_one__le__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X2 ) )
          = ( ord_less_eq_real @ A @ X2 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_9087_log__le__zero__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A @ X2 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_9088_zero__le__log__cancel__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X2 ) )
          = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_9089_cot__npi,axiom,
    ! [N2: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9090_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_9091_cot__periodic,axiom,
    ! [X2: real] :
      ( ( cot_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X2 ) ) ).

% cot_periodic
thf(fact_9092_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9093_sinh__less__cosh__real,axiom,
    ! [X2: real] : ( ord_less_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).

% sinh_less_cosh_real
thf(fact_9094_sinh__le__cosh__real,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).

% sinh_le_cosh_real
thf(fact_9095_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
thf(fact_9096_log__def,axiom,
    ( log
    = ( ^ [A4: real,X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ A4 ) ) ) ) ).

% log_def
thf(fact_9097_cosh__real__pos,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_pos
thf(fact_9098_cosh__real__nonpos__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9099_cosh__real__nonneg__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9100_cosh__real__nonneg,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_nonneg
thf(fact_9101_cosh__real__ge__1,axiom,
    ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_ge_1
thf(fact_9102_cosh__real__strict__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9103_cosh__real__nonneg__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_real @ X2 @ Y2 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9104_cosh__real__nonpos__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_real @ Y2 @ X2 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9105_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_9106_arcosh__cosh__real,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( arcosh_real @ ( cosh_real @ X2 ) )
        = X2 ) ) ).

% arcosh_cosh_real
thf(fact_9107_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9108_Complex__eq__i,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( complex2 @ X2 @ Y2 )
        = imaginary_unit )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9109_log__base__change,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X2 )
          = ( divide_divide_real @ ( log @ A @ X2 ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_9110_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_9111_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_9112_log__mult,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y2 )
           => ( ( log @ A @ ( times_times_real @ X2 @ Y2 ) )
              = ( plus_plus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_9113_log__divide,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y2 )
           => ( ( log @ A @ ( divide_divide_real @ X2 @ Y2 ) )
              = ( minus_minus_real @ ( log @ A @ X2 ) @ ( log @ A @ Y2 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_9114_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_9115_log__base__pow,axiom,
    ! [A: real,N2: nat,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N2 ) @ X2 )
        = ( divide_divide_real @ ( log @ A @ X2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_9116_log__nat__power,axiom,
    ! [X2: real,B: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ B @ ( power_power_real @ X2 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X2 ) ) ) ) ).

% log_nat_power
thf(fact_9117_log__inverse,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( log @ A @ ( inverse_inverse_real @ X2 ) )
            = ( uminus_uminus_real @ ( log @ A @ X2 ) ) ) ) ) ) ).

% log_inverse
thf(fact_9118_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_9119_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_9120_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X2: 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 @ X2 )
             => ( ( log @ A @ X2 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X2 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_9121_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_9122_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_9123_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_9124_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_9125_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_9126_cosh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9127_cot__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).

% cot_gt_zero
thf(fact_9128_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_9129_log__base__10__eq2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% log_base_10_eq2
thf(fact_9130_sinh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9131_tan__cot_H,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
      = ( cot_real @ X2 ) ) ).

% tan_cot'
thf(fact_9132_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_9133_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_9134_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9135_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_9136_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_9137_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_9138_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_9139_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_9140_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_9141_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X2 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
            & ( ord_less_eq_real @ X2 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_9142_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_9143_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_9144_powr__gt__zero,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X2 @ A ) )
      = ( X2 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_9145_powr__nonneg__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X2 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_9146_powr__less__cancel__iff,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_9147_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_9148_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9149_powr__eq__one__iff,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X2 )
          = one_one_real )
        = ( X2 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_9150_powr__one__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( powr_real @ X2 @ one_one_real )
        = X2 )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% powr_one_gt_zero_iff
thf(fact_9151_powr__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ one_one_real )
        = X2 ) ) ).

% powr_one
thf(fact_9152_powr__le__cancel__iff,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_9153_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_9154_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cis_pi
thf(fact_9155_powr__log__cancel,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( powr_real @ A @ ( log @ A @ X2 ) )
            = X2 ) ) ) ) ).

% powr_log_cancel
thf(fact_9156_log__powr__cancel,axiom,
    ! [A: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y2 ) )
          = Y2 ) ) ) ).

% log_powr_cancel
thf(fact_9157_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_9158_powr__numeral,axiom,
    ! [X2: real,N2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N2 ) )
        = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% powr_numeral
thf(fact_9159_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9160_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_9161_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9162_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_9163_square__powr__half,axiom,
    ! [X2: real] :
      ( ( powr_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X2 ) ) ).

% square_powr_half
thf(fact_9164_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_9165_powr__powr,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X2 @ A ) @ B )
      = ( powr_real @ X2 @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_9166_powr__non__neg,axiom,
    ! [A: real,X2: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X2 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_9167_powr__less__mono2__neg,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y2 )
         => ( ord_less_real @ ( powr_real @ Y2 @ A ) @ ( powr_real @ X2 @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_9168_powr__ge__pzero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y2 ) ) ).

% powr_ge_pzero
thf(fact_9169_powr__mono2,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y2 )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_9170_powr__less__mono,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).

% powr_less_mono
thf(fact_9171_powr__less__cancel,axiom,
    ! [X2: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_9172_powr__mono,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X2 )
       => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ X2 @ B ) ) ) ) ).

% powr_mono
thf(fact_9173_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9174_powr__mono2_H,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y2 )
         => ( ord_less_eq_real @ ( powr_real @ Y2 @ A ) @ ( powr_real @ X2 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_9175_powr__less__mono2,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y2 )
         => ( ord_less_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_9176_gr__one__powr,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y2 ) ) ) ) ).

% gr_one_powr
thf(fact_9177_powr__inj,axiom,
    ! [A: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X2 )
            = ( powr_real @ A @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% powr_inj
thf(fact_9178_ge__one__powr__ge__zero,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_9179_powr__mono__both,axiom,
    ! [A: real,B: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ Y2 )
           => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_9180_powr__le1,axiom,
    ! [A: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_9181_powr__divide,axiom,
    ! [X2: real,Y2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( powr_real @ ( divide_divide_real @ X2 @ Y2 ) @ A )
          = ( divide_divide_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_divide
thf(fact_9182_powr__mult,axiom,
    ! [X2: real,Y2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( powr_real @ ( times_times_real @ X2 @ Y2 ) @ A )
          = ( times_times_real @ ( powr_real @ X2 @ A ) @ ( powr_real @ Y2 @ A ) ) ) ) ) ).

% powr_mult
thf(fact_9183_inverse__powr,axiom,
    ! [Y2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( powr_real @ ( inverse_inverse_real @ Y2 ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y2 @ A ) ) ) ) ).

% inverse_powr
thf(fact_9184_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_9185_log__base__powr,axiom,
    ! [A: real,B: real,X2: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X2 )
        = ( divide_divide_real @ ( log @ A @ X2 ) @ B ) ) ) ).

% log_base_powr
thf(fact_9186_ln__powr,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X2 @ Y2 ) )
        = ( times_times_real @ Y2 @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_powr
thf(fact_9187_log__powr,axiom,
    ! [X2: real,B: real,Y2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X2 @ Y2 ) )
        = ( times_times_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ).

% log_powr
thf(fact_9188_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X2 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X2 )
            & ( ord_less_real @ X2 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9189_powr__realpow,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) )
        = ( power_power_real @ X2 @ N2 ) ) ) ).

% powr_realpow
thf(fact_9190_less__log__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ Y2 @ ( log @ B @ X2 ) )
          = ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ).

% less_log_iff
thf(fact_9191_log__less__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ B @ X2 ) @ Y2 )
          = ( ord_less_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ).

% log_less_iff
thf(fact_9192_less__powr__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ ( powr_real @ B @ Y2 ) )
          = ( ord_less_real @ ( log @ B @ X2 ) @ Y2 ) ) ) ) ).

% less_powr_iff
thf(fact_9193_powr__less__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( powr_real @ B @ Y2 ) @ X2 )
          = ( ord_less_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ) ).

% powr_less_iff
thf(fact_9194_floor__eq,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = N2 ) ) ) ).

% floor_eq
thf(fact_9195_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_9196_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_9197_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_9198_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_9199_DeMoivre,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N2 )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) ) ) ).

% DeMoivre
thf(fact_9200_powr__neg__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).

% powr_neg_one
thf(fact_9201_powr__mult__base,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y2 ) )
        = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y2 ) ) ) ) ).

% powr_mult_base
thf(fact_9202_powr__le__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X2 )
          = ( ord_less_eq_real @ Y2 @ ( log @ B @ X2 ) ) ) ) ) ).

% powr_le_iff
thf(fact_9203_le__powr__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y2 ) )
          = ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y2 ) ) ) ) ).

% le_powr_iff
thf(fact_9204_log__le__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ B @ X2 ) @ Y2 )
          = ( ord_less_eq_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ).

% log_le_iff
thf(fact_9205_le__log__iff,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ Y2 @ ( log @ B @ X2 ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ).

% le_log_iff
thf(fact_9206_floor__eq2,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_9207_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_9208_ln__powr__bound,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_9209_ln__powr__bound2,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X2 ) ) ) ) ).

% ln_powr_bound2
thf(fact_9210_add__log__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ Y2 @ ( log @ B @ X2 ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_9211_log__add__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ ( log @ B @ X2 ) @ Y2 )
            = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ Y2 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_9212_minus__log__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ Y2 @ ( log @ B @ X2 ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y2 ) @ X2 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_9213_log__minus__eq__powr,axiom,
    ! [B: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ ( log @ B @ X2 ) @ Y2 )
            = ( log @ B @ ( times_times_real @ X2 @ ( powr_real @ B @ ( uminus_uminus_real @ Y2 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_9214_powr__half__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X2 ) ) ) ).

% powr_half_sqrt
thf(fact_9215_powr__neg__numeral,axiom,
    ! [X2: real,N2: num] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_9216_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_9217_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9218_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9219_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9220_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_9221_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_9222_complex__of__real__mult__Complex,axiom,
    ! [R: real,X2: real,Y2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X2 @ Y2 ) )
      = ( complex2 @ ( times_times_real @ R @ X2 ) @ ( times_times_real @ R @ Y2 ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9223_Complex__mult__complex__of__real,axiom,
    ! [X2: real,Y2: real,R: real] :
      ( ( times_times_complex @ ( complex2 @ X2 @ Y2 ) @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ ( times_times_real @ X2 @ R ) @ ( times_times_real @ Y2 @ R ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9224_Complex__add__complex__of__real,axiom,
    ! [X2: real,Y2: real,R: real] :
      ( ( plus_plus_complex @ ( complex2 @ X2 @ Y2 ) @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ ( plus_plus_real @ X2 @ R ) @ Y2 ) ) ).

% Complex_add_complex_of_real
thf(fact_9225_complex__of__real__add__Complex,axiom,
    ! [R: real,X2: real,Y2: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X2 @ Y2 ) )
      = ( complex2 @ ( plus_plus_real @ R @ X2 ) @ Y2 ) ) ).

% complex_of_real_add_Complex
thf(fact_9226_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_9227_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_9228_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [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 ) ) ) )
              & ( ( tan_real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_9229_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [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 ) ) ) )
              & ( ( sin_real @ X )
                = Y ) ) ) ) ) ).

% arcsin_def
thf(fact_9230_modulo__int__unfold,axiom,
    ! [L2: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = 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 @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 )
                @ ( 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_9231_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% csqrt_eq_1
thf(fact_9232_csqrt__1,axiom,
    ( ( csqrt @ one_one_complex )
    = one_one_complex ) ).

% csqrt_1
thf(fact_9233_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9234_sgn__mod,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L2 @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
          = ( sgn_sgn_int @ L2 ) ) ) ) ).

% sgn_mod
thf(fact_9235_ln__neg__is__const,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ln_ln_real @ X2 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9236_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9237_of__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).

% of_real_sqrt
thf(fact_9238_arccos__def,axiom,
    ( arccos
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y ) ) ) ) ) ).

% arccos_def
thf(fact_9239_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9240_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9241_divide__int__unfold,axiom,
    ! [L2: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = 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 @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( 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_9242_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
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9243_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9244_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9245_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_9246_sgn__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9247_zero__le__sgn__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_sgn_iff
thf(fact_9248_real__root__zero,axiom,
    ! [N2: nat] :
      ( ( root @ N2 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9249_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9250_real__root__Suc__0,axiom,
    ! [X2: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
      = X2 ) ).

% real_root_Suc_0
thf(fact_9251_real__root__eq__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = ( root @ N2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% real_root_eq_iff
thf(fact_9252_root__0,axiom,
    ! [X2: real] :
      ( ( root @ zero_zero_nat @ X2 )
      = zero_zero_real ) ).

% root_0
thf(fact_9253_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9254_real__root__eq__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = zero_zero_real )
        = ( X2 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9255_real__root__less__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_real @ X2 @ Y2 ) ) ) ).

% real_root_less_iff
thf(fact_9256_real__root__le__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% real_root_le_iff
thf(fact_9257_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_9258_real__root__eq__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = one_one_real )
        = ( X2 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9259_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_9260_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_real @ zero_zero_real @ Y2 ) ) ) ).

% real_root_gt_0_iff
thf(fact_9261_real__root__lt__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X2 ) @ zero_zero_real )
        = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9262_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) ).

% real_root_ge_0_iff
thf(fact_9263_real__root__le__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9264_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_9265_real__root__lt__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X2 ) @ one_one_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9266_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_real @ one_one_real @ Y2 ) ) ) ).

% real_root_gt_1_iff
thf(fact_9267_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq_real @ one_one_real @ Y2 ) ) ) ).

% real_root_ge_1_iff
thf(fact_9268_real__root__le__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ one_one_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9269_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_9270_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = ( nat2 @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9271_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( nat2 @ Y2 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9272_nat__ceiling__le__eq,axiom,
    ! [X2: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A )
      = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9273_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_9274_real__root__pow__pos2,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( root @ N2 @ X2 ) @ N2 )
          = X2 ) ) ) ).

% real_root_pow_pos2
thf(fact_9275_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_9276_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9277_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9278_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9279_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X2: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9280_real__root__divide,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( root @ N2 @ ( divide_divide_real @ X2 @ Y2 ) )
      = ( divide_divide_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ).

% real_root_divide
thf(fact_9281_sgn__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X2 ) )
        = ( sgn_sgn_real @ X2 ) ) ) ).

% sgn_root
thf(fact_9282_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_9283_real__root__commute,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( root @ M @ ( root @ N2 @ X2 ) )
      = ( root @ N2 @ ( root @ M @ X2 ) ) ) ).

% real_root_commute
thf(fact_9284_nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% nat_mask_eq
thf(fact_9285_real__root__mult,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( root @ N2 @ ( times_times_real @ X2 @ Y2 ) )
      = ( times_times_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ).

% real_root_mult
thf(fact_9286_real__root__mult__exp,axiom,
    ! [M: nat,N2: nat,X2: real] :
      ( ( root @ ( times_times_nat @ M @ N2 ) @ X2 )
      = ( root @ M @ ( root @ N2 @ X2 ) ) ) ).

% real_root_mult_exp
thf(fact_9287_real__root__minus,axiom,
    ! [N2: nat,X2: real] :
      ( ( root @ N2 @ ( uminus_uminus_real @ X2 ) )
      = ( uminus_uminus_real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_minus
thf(fact_9288_real__root__inverse,axiom,
    ! [N2: nat,X2: real] :
      ( ( root @ N2 @ ( inverse_inverse_real @ X2 ) )
      = ( inverse_inverse_real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_inverse
thf(fact_9289_real__root__pos__pos__le,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_pos_pos_le
thf(fact_9290_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9291_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( divide_divide_real @ X @ ( abs_abs_real @ X ) ) ) ) ).

% real_sgn_eq
thf(fact_9292_nat__mono,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ).

% nat_mono
thf(fact_9293_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9294_root__sgn__power,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) ) )
        = Y2 ) ) ).

% root_sgn_power
thf(fact_9295_sgn__power__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X2 ) ) @ N2 ) )
        = X2 ) ) ).

% sgn_power_root
thf(fact_9296_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9297_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_9298_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9299_split__root,axiom,
    ! [P: real > $o,N2: nat,X2: real] :
      ( ( P @ ( root @ N2 @ X2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
                = X2 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_9300_real__root__less__mono,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ) ).

% real_root_less_mono
thf(fact_9301_real__root__le__mono,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ) ).

% real_root_le_mono
thf(fact_9302_real__root__power,axiom,
    ! [N2: nat,X2: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X2 @ K ) )
        = ( power_power_real @ ( root @ N2 @ X2 ) @ K ) ) ) ).

% real_root_power
thf(fact_9303_real__root__abs,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X2 ) )
        = ( abs_abs_real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_abs
thf(fact_9304_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_9305_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_9306_nat__le__iff,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N2 )
      = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_9307_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_9308_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_9309_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9310_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9311_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% or_nat_def
thf(fact_9312_real__nat__ceiling__ge,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9313_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_9314_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9315_real__root__gt__zero,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_gt_zero
thf(fact_9316_real__root__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N3 )
       => ( ( ord_less_real @ one_one_real @ X2 )
         => ( ord_less_real @ ( root @ N3 @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9317_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9318_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9319_root__abs__power,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y2 @ N2 ) ) )
        = ( abs_abs_real @ Y2 ) ) ) ).

% root_abs_power
thf(fact_9320_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_9321_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_9322_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_9323_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9324_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9325_Suc__as__int,axiom,
    ( suc
    = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9326_nat__abs__triangle__ineq,axiom,
    ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9327_nat__div__distrib,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( nat2 @ ( divide_divide_int @ X2 @ Y2 ) )
        = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).

% nat_div_distrib
thf(fact_9328_nat__div__distrib_H,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( nat2 @ ( divide_divide_int @ X2 @ Y2 ) )
        = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).

% nat_div_distrib'
thf(fact_9329_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_9330_nat__floor__neg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9331_nat__mod__distrib,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y2 ) )
          = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9332_div__abs__eq__div__nat,axiom,
    ! [K: int,L2: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_9333_floor__eq3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_9334_le__nat__floor,axiom,
    ! [X2: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A )
     => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9335_real__root__pos__pos,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_pos_pos
thf(fact_9336_real__root__strict__increasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N3 )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ X2 @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N3 @ X2 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9337_real__root__decreasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N3 )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ord_less_eq_real @ ( root @ N3 @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9338_real__root__pow__pos,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( root @ N2 @ X2 ) @ N2 )
          = X2 ) ) ) ).

% real_root_pow_pos
thf(fact_9339_odd__real__root__power__cancel,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X2 @ N2 ) )
        = X2 ) ) ).

% odd_real_root_power_cancel
thf(fact_9340_odd__real__root__unique,axiom,
    ! [N2: nat,Y2: real,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ( power_power_real @ Y2 @ N2 )
          = X2 )
       => ( ( root @ N2 @ X2 )
          = Y2 ) ) ) ).

% odd_real_root_unique
thf(fact_9341_odd__real__root__pow,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( root @ N2 @ X2 ) @ N2 )
        = X2 ) ) ).

% odd_real_root_pow
thf(fact_9342_real__root__pos__unique,axiom,
    ! [N2: nat,Y2: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( power_power_real @ Y2 @ N2 )
            = X2 )
         => ( ( root @ N2 @ X2 )
            = Y2 ) ) ) ) ).

% real_root_pos_unique
thf(fact_9343_real__root__power__cancel,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( root @ N2 @ ( power_power_real @ X2 @ N2 ) )
          = X2 ) ) ) ).

% real_root_power_cancel
thf(fact_9344_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9345_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X2: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X2 )
     => ( ( X2
          = ( 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_9346_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_9347_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_9348_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9349_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_9350_floor__eq4,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_9351_real__root__increasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N3 @ X2 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9352_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_9353_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_9354_cis__Arg__unique,axiom,
    ! [Z: complex,X2: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X2 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ( arg @ Z )
            = X2 ) ) ) ) ).

% cis_Arg_unique
thf(fact_9355_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_9356_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_9357_log__base__root,axiom,
    ! [N2: nat,B: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X2 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X2 ) ) ) ) ) ).

% log_base_root
thf(fact_9358_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9359_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_9360_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9361_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_9362_root__powr__inverse,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( root @ N2 @ X2 )
          = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9363_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_9364_powr__real__of__int,axiom,
    ! [X2: real,N2: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N2 ) )
            = ( power_power_real @ X2 @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N2 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9365_arctan__inverse,axiom,
    ! [X2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X2 ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X2 ) ) ) ) ).

% arctan_inverse
thf(fact_9366_powr__int,axiom,
    ! [X2: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X2 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9367_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9368_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_9369_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_9370_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_9371_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_9372_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_9373_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_9374_take__bit__mult,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).

% take_bit_mult
thf(fact_9375_take__bit__diff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% take_bit_diff
thf(fact_9376_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9377_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_9378_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_9379_concat__bit__eq__iff,axiom,
    ! [N2: nat,K: int,L2: int,R: int,S3: int] :
      ( ( ( bit_concat_bit @ N2 @ K @ L2 )
        = ( bit_concat_bit @ N2 @ R @ S3 ) )
      = ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = ( bit_se2923211474154528505it_int @ N2 @ R ) )
        & ( L2 = S3 ) ) ) ).

% concat_bit_eq_iff
thf(fact_9380_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_9381_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_9382_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_9383_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_9384_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_9385_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_9386_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_9387_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_9388_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_9389_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_9390_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_nat_def
thf(fact_9391_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_9392_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_def
thf(fact_9393_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9394_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_9395_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_9396_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_9397_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_9398_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_9399_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_9400_take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9401_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_9402_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_9403_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_9404_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_9405_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9406_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_9407_take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9408_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_9409_Arg__def,axiom,
    ( arg
    = ( ^ [Z5: complex] :
          ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z5 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9410_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L
            @ ( if_int
              @ ( L
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9411_and__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9412_and__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9413_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9414_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_9415_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_9416_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_9417_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_9418_num__induct,axiom,
    ! [P: num > $o,X2: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X2 ) ) ) ).

% num_induct
thf(fact_9419_add__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( plus_plus_num @ X2 @ ( inc @ Y2 ) )
      = ( inc @ ( plus_plus_num @ X2 @ Y2 ) ) ) ).

% add_inc
thf(fact_9420_AND__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) ) ) ).

% AND_lower
thf(fact_9421_AND__upper1,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ X2 ) ) ).

% AND_upper1
thf(fact_9422_AND__upper2,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ Y2 ) ) ).

% AND_upper2
thf(fact_9423_AND__upper1_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y2 @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9424_AND__upper2_H,axiom,
    ! [Y2: int,Z: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9425_plus__and__or,axiom,
    ! [X2: int,Y2: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ ( bit_se1409905431419307370or_int @ X2 @ Y2 ) )
      = ( plus_plus_int @ X2 @ Y2 ) ) ).

% plus_and_or
thf(fact_9426_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9427_inc_Osimps_I2_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit0 @ X2 ) )
      = ( bit1 @ X2 ) ) ).

% inc.simps(2)
thf(fact_9428_inc_Osimps_I3_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit1 @ X2 ) )
      = ( bit0 @ ( inc @ X2 ) ) ) ).

% inc.simps(3)
thf(fact_9429_add__One,axiom,
    ! [X2: num] :
      ( ( plus_plus_num @ X2 @ one )
      = ( inc @ X2 ) ) ).

% add_One
thf(fact_9430_and__less__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).

% and_less_eq
thf(fact_9431_AND__upper1_H_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y2 @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9432_AND__upper2_H_H,axiom,
    ! [Y2: int,Z: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9433_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_9434_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_9435_mult__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( times_times_num @ X2 @ ( inc @ Y2 ) )
      = ( plus_plus_num @ ( times_times_num @ X2 @ Y2 ) @ X2 ) ) ).

% mult_inc
thf(fact_9436_even__and__iff__int,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).

% even_and_iff_int
thf(fact_9437_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9438_and__int_Oelims,axiom,
    ! [X2: int,Xa2: int,Y2: int] :
      ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( ( member_int @ X2 @ ( 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 ) ) ) )
         => ( Y2
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X2 @ ( 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 ) ) ) )
         => ( Y2
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9439_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( 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 ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9440_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9441_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_9442_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_9443_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_9444_and__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_9445_and__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_9446_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_9447_and__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_9448_and__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_9449_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_9450_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_9451_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_9452_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_9453_bit__and__int__iff,axiom,
    ! [K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        & ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).

% bit_and_int_iff
thf(fact_9454_bit__or__int__iff,axiom,
    ! [K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        | ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).

% bit_or_int_iff
thf(fact_9455_pow_Osimps_I1_J,axiom,
    ! [X2: num] :
      ( ( pow @ X2 @ one )
      = X2 ) ).

% pow.simps(1)
thf(fact_9456_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_9457_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% and_nat_def
thf(fact_9458_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_9459_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9460_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_9461_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_9462_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( bit_concat_bit @ N @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9463_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N4 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9464_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_int_def
thf(fact_9465_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N: nat] :
          ( if_nat
          @ ( ( M6 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9466_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9467_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% set_bit_eq
thf(fact_9468_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9469_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_9470_and__int_Opsimps,axiom,
    ! [K: int,L2: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
     => ( ( ( ( member_int @ K @ ( 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 ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( 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 ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9471_and__int_Opelims,axiom,
    ! [X2: int,Xa2: int,Y2: int] :
      ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X2 @ ( 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 ) ) ) )
               => ( Y2
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X2 @ ( 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 ) ) ) )
               => ( Y2
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( 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 @ X2 @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_9472_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_9473_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_9474_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_9475_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9476_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_9477_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9478_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_9479_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_9480_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_9481_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_9482_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_9483_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_9484_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_9485_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_9486_sin__times__pi__eq__0,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ ( times_times_real @ X2 @ pi ) )
        = zero_zero_real )
      = ( member_real @ X2 @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9487_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_9488_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M6: nat,N: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_nat_def
thf(fact_9489_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_9490_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_9491_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_9492_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( 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 @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_9493_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I4: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I4 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I4 @ J2 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) )
             => ( P @ I4 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_9494_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_9495_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_9496_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_9497_xor__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9498_xor__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9499_xor__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9500_xor__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9501_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9502_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9503_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X5: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( ( ord_less_eq_nat @ M9 @ M6 )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M9 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X5 @ M6 ) @ ( X5 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9504_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_9505_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_9506_concat__bit__of__zero__1,axiom,
    ! [N2: nat,L2: int] :
      ( ( bit_concat_bit @ N2 @ zero_zero_int @ L2 )
      = ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ).

% concat_bit_of_zero_1
thf(fact_9507_xor__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9508_xor__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9509_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_9510_bit__xor__int__iff,axiom,
    ! [K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
       != ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).

% bit_xor_int_iff
thf(fact_9511_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9512_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_9513_XOR__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y2 ) ) ) ) ).

% XOR_lower
thf(fact_9514_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M6: nat,N: nat] : ( bit_se1412395901928357646or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9515_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M6: nat,N: nat] : ( bit_se6528837805403552850or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9516_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_9517_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% xor_nat_def
thf(fact_9518_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9519_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).

% concat_bit_eq
thf(fact_9520_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).

% concat_bit_def
thf(fact_9521_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9522_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_nat_def
thf(fact_9523_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_int_def
thf(fact_9524_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_9525_XOR__upper,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_9526_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9527_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L )
          @ ( if_int
            @ ( L
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9528_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_9529_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_9530_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_9531_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_9532_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_9533_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9534_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_9535_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_9536_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_9537_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9538_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9539_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9540_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).

% or_int_def
thf(fact_9541_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9542_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_9543_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_9544_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_9545_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9546_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).

% xor_int_def
thf(fact_9547_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_9548_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_9549_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_9550_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_9551_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_9552_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_9553_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_9554_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_9555_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_9556_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_9557_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_9558_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9559_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_9560_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_9561_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_9562_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_9563_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_9564_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_9565_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_9566_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_9567_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_9568_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_9569_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_9570_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_9571_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_9572_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_9573_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_9574_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9575_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I4: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J2 ) ) ) )
     => ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I4 ) ) ) )
       => ( 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_9576_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9577_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_9578_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9579_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9580_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9581_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_9582_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_9583_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_9584_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R4 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9585_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_9586_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_9587_Re__divide__of__real,axiom,
    ! [Z: complex,R: real] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ R ) ) ).

% Re_divide_of_real
thf(fact_9588_Re__sgn,axiom,
    ! [Z: complex] :
      ( ( re @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Re_sgn
thf(fact_9589_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_9590_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9591_complex__Re__le__cmod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( re @ X2 ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% complex_Re_le_cmod
thf(fact_9592_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_9593_plus__complex_Osimps_I1_J,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( re @ ( plus_plus_complex @ X2 @ Y2 ) )
      = ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) ) ).

% plus_complex.simps(1)
thf(fact_9594_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X2: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R @ X2 ) )
      = ( times_times_real @ R @ ( re @ X2 ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9595_abs__Re__le__cmod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% abs_Re_le_cmod
thf(fact_9596_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9597_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_9598_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9599_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_9600_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_9601_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z5: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z5 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9602_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_9603_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9604_Im__divide__of__real,axiom,
    ! [Z: complex,R: real] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ R ) ) ).

% Im_divide_of_real
thf(fact_9605_Im__sgn,axiom,
    ! [Z: complex] :
      ( ( im @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Im_sgn
thf(fact_9606_Re__power__real,axiom,
    ! [X2: complex,N2: nat] :
      ( ( ( im @ X2 )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X2 @ N2 ) )
        = ( power_power_real @ ( re @ X2 ) @ N2 ) ) ) ).

% Re_power_real
thf(fact_9607_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_9608_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_9609_csqrt__of__real__nonneg,axiom,
    ! [X2: complex] :
      ( ( ( im @ X2 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) )
       => ( ( csqrt @ X2 )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X2 ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9610_csqrt__minus,axiom,
    ! [X2: complex] :
      ( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
        | ( ( ( im @ X2 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).

% csqrt_minus
thf(fact_9611_csqrt__of__real__nonpos,axiom,
    ! [X2: complex] :
      ( ( ( im @ X2 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
       => ( ( csqrt @ X2 )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9612_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_9613_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_9614_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_9615_plus__complex_Osimps_I2_J,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( im @ ( plus_plus_complex @ X2 @ Y2 ) )
      = ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ).

% plus_complex.simps(2)
thf(fact_9616_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X2: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R @ X2 ) )
      = ( times_times_real @ R @ ( im @ X2 ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9617_abs__Im__le__cmod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% abs_Im_le_cmod
thf(fact_9618_times__complex_Osimps_I2_J,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( im @ ( times_times_complex @ X2 @ Y2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9619_cmod__Im__le__iff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( re @ X2 )
        = ( re @ Y2 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( abs_abs_real @ ( im @ Y2 ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_9620_cmod__Re__le__iff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( im @ X2 )
        = ( im @ Y2 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( abs_abs_real @ ( re @ Y2 ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_9621_times__complex_Osimps_I1_J,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( re @ ( times_times_complex @ X2 @ Y2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9622_plus__complex_Ocode,axiom,
    ( plus_plus_complex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ) ).

% plus_complex.code
thf(fact_9623_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R4: real,X: complex] : ( complex2 @ ( times_times_real @ R4 @ ( re @ X ) ) @ ( times_times_real @ R4 @ ( im @ X ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9624_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_9625_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_9626_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_9627_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_9628_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_9629_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9630_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_9631_Im__power2,axiom,
    ! [X2: complex] :
      ( ( im @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X2 ) ) @ ( im @ X2 ) ) ) ).

% Im_power2
thf(fact_9632_Re__power2,axiom,
    ! [X2: complex] :
      ( ( re @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9633_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_9634_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9635_inverse__complex_Osimps_I1_J,axiom,
    ! [X2: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X2 ) )
      = ( 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 ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9636_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_9637_Re__divide,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X2 @ Y2 ) )
      = ( 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 ) ) ) ) ) ) ).

% Re_divide
thf(fact_9638_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_9639_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_9640_inverse__complex_Osimps_I2_J,axiom,
    ! [X2: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X2 ) )
      = ( 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.simps(2)
thf(fact_9641_Im__divide,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X2 @ Y2 ) )
      = ( 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 ) ) ) ) ) ) ).

% Im_divide
thf(fact_9642_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_9643_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_9644_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X: complex] : ( complex2 @ ( 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 ) ) ) ) ) @ ( 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.code
thf(fact_9645_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( 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 ) ) ) ) ) @ ( 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 ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9646_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( ( member_complex @ R @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9647_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( ( member_complex @ R @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9648_Re__divide__Reals,axiom,
    ! [R: complex,Z: complex] :
      ( ( member_complex @ R @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ Z @ R ) )
        = ( divide_divide_real @ ( re @ Z ) @ ( re @ R ) ) ) ) ).

% Re_divide_Reals
thf(fact_9649_Im__divide__Reals,axiom,
    ! [R: complex,Z: complex] :
      ( ( member_complex @ R @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ Z @ R ) )
        = ( divide_divide_real @ ( im @ Z ) @ ( re @ R ) ) ) ) ).

% Im_divide_Reals
thf(fact_9650_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_9651_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_9652_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( ( cnj @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
thf(fact_9653_complex__cnj__one,axiom,
    ( ( cnj @ one_one_complex )
    = one_one_complex ) ).

% complex_cnj_one
thf(fact_9654_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9655_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_9656_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_9657_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_9658_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_9659_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_9660_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_9661_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_9662_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_9663_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_9664_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_9665_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_9666_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9667_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_9668_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_9669_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_9670_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9671_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_9672_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9673_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9674_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_9675_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9676_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_9677_int__of__integer__max,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( code_int_of_integer @ ( ord_max_Code_integer @ K @ L2 ) )
      = ( ord_max_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L2 ) ) ) ).

% int_of_integer_max
thf(fact_9678_card__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).

% card_atLeastAtMost
thf(fact_9679_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_9680_card__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_9681_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9682_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9683_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9684_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_9685_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9686_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S ) ) ).

% card_sum_le_nat_sum
thf(fact_9687_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z5: complex] :
                  ( ( power_power_complex @ Z5 @ N2 )
                  = C ) ) )
          = N2 ) ) ) ).

% card_nth_roots
thf(fact_9688_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) ) )
        = N2 ) ) ).

% card_roots_unity_eq
thf(fact_9689_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9690_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_9691_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_9692_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_9693_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_9694_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_9695_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_9696_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N2 @ M3 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9697_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9698_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9699_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9700_bezw__0,axiom,
    ! [X2: nat] :
      ( ( bezw @ X2 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9701_drop__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9702_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_9703_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_9704_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_9705_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_9706_drop__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9707_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_9708_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_9709_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_int_def
thf(fact_9710_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_9711_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_9712_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_9713_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9714_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9715_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9716_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9717_prod__decode__aux_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y2: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
         => ( Y2
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
         => ( Y2
            = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9718_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9719_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R )
     => ~ ! [S2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S2 )
           => ! [T4: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T4 )
               => ( R
                 != ( plus_plus_rat @ S2 @ T4 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9720_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_9721_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_9722_bezw_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y2: product_prod_int_int] :
      ( ( ( bezw @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y2
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y2
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9723_bezw__non__0,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y2 )
     => ( ( bezw @ X2 @ Y2 )
        = ( 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_non_0
thf(fact_9724_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( 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.simps
thf(fact_9725_bezw_Opelims,axiom,
    ! [X2: nat,Xa2: nat,Y2: product_prod_int_int] :
      ( ( ( bezw @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y2
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y2
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9726_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_9727_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_9728_prod__decode__aux_Opelims,axiom,
    ! [X2: nat,Xa2: nat,Y2: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
               => ( Y2
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
               => ( Y2
                  = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9729_rat__inverse__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B4 ) @ ( abs_abs_int @ A4 ) ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_inverse_code
thf(fact_9730_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_9731_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9732_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9733_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_9734_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_9735_rat__sgn__code,axiom,
    ! [P6: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P6 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P6 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9736_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9737_vebt__maxt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y2
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y2
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y2 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = 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] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( 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_9738_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K2 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9739_vebt__mint_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
       => ( ! [A3: $o,B3: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y2
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y2
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y2 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = 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] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y2
                      = ( 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_9740_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y2
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y2
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y2
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y2
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y2
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9741_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_9742_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_9743_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_9744_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ! [Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9745_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( 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_9746_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R4: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R4 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9747_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9748_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
              @ ( produc6916734918728496179nteger
                @ ^ [R4: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S5 ) ) )
                @ ( code_divmod_abs @ K3 @ L ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R4: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S5 ) ) )
                    @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9749_card__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_9750_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M6: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M6 @ N )
            & ( member_nat @ N @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_9751_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9752_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M5: nat] :
          ( ( ord_less_nat @ K @ M5 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M5 @ N9 )
              & ( member_nat @ N9 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_9753_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M6: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M6 @ N )
            & ( member_nat @ N @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9754_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S ) )
             => ( member_nat @ ( R3 @ N9 ) @ S ) ) ) ) ).

% finite_enumerate
thf(fact_9755_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_9756_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_9757_card__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).

% card_greaterThanLessThan
thf(fact_9758_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
      = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9759_tanh__real__bounds,axiom,
    ! [X2: real] : ( member_real @ ( tanh_real @ X2 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9760_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_9761_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9762_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L ) )
                @ ( code_divmod_abs @ K3 @ L )
                @ ( produc6916734918728496179nteger
                  @ ^ [R4: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S5 ) ) )
                  @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9763_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9764_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_9765_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
          @ Xa2
          @ X2 ) ) ) ).

% times_int.abs_eq
thf(fact_9766_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y5: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y5 ) ) ) ).

% eq_Abs_Integ
thf(fact_9767_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9768_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9769_uminus__int_Oabs__eq,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X2 ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
          @ X2 ) ) ) ).

% uminus_int.abs_eq
thf(fact_9770_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9771_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X2 ) ) ).

% less_int.abs_eq
thf(fact_9772_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X2 ) ) ).

% less_eq_int.abs_eq
thf(fact_9773_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
          @ Xa2
          @ X2 ) ) ) ).

% plus_int.abs_eq
thf(fact_9774_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
          @ Xa2
          @ X2 ) ) ) ).

% minus_int.abs_eq
thf(fact_9775_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_9776_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9777_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9778_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_9779_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_9780_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_9781_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_9782_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9783_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9784_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_9785_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% uminus_int_def
thf(fact_9786_pow_Osimps_I3_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( pow @ X2 @ ( bit1 @ Y2 ) )
      = ( times_times_num @ ( sqr @ ( pow @ X2 @ Y2 ) ) @ X2 ) ) ).

% pow.simps(3)
thf(fact_9787_sqr_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N2 ) ) ) ) ).

% sqr.simps(2)
thf(fact_9788_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9789_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).

% sqr_conv_mult
thf(fact_9790_pow_Osimps_I2_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( pow @ X2 @ ( bit0 @ Y2 ) )
      = ( sqr @ ( pow @ X2 @ Y2 ) ) ) ).

% pow.simps(2)
thf(fact_9791_sqr_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).

% sqr.simps(3)
thf(fact_9792_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9793_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9794_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9795_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_9796_normalize__denom__zero,axiom,
    ! [P6: int] :
      ( ( normalize @ ( product_Pair_int_int @ P6 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9797_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
      = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9798_normalize__def,axiom,
    ( normalize
    = ( ^ [P4: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P4 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9799_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_9800_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y2: nat,X2: nat] :
      ( ( ( ord_less_nat @ C @ Y2 )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y2 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y2 )
       => ( ( ( ord_less_nat @ X2 @ Y2 )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X2 @ Y2 )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9801_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9802_bij__betw__Suc,axiom,
    ! [M7: set_nat,N3: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N3 ) ) ).

% bij_betw_Suc
thf(fact_9803_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_9804_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_9805_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9806_eq__rat_I2_J,axiom,
    ! [A: int] :
      ( ( fract @ A @ zero_zero_int )
      = ( fract @ zero_zero_int @ one_one_int ) ) ).

% eq_rat(2)
thf(fact_9807_Fract__of__nat__eq,axiom,
    ! [K: nat] :
      ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
      = ( semiri681578069525770553at_rat @ K ) ) ).

% Fract_of_nat_eq
thf(fact_9808_One__rat__def,axiom,
    ( one_one_rat
    = ( fract @ one_one_int @ one_one_int ) ) ).

% One_rat_def
thf(fact_9809_Fract__of__int__eq,axiom,
    ! [K: int] :
      ( ( fract @ K @ one_one_int )
      = ( ring_1_of_int_rat @ K ) ) ).

% Fract_of_int_eq
thf(fact_9810_Zero__rat__def,axiom,
    ( zero_zero_rat
    = ( fract @ zero_zero_int @ one_one_int ) ) ).

% Zero_rat_def
thf(fact_9811_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_9812_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_9813_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9814_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_9815_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_9816_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_9817_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_9818_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_9819_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_9820_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_int @ A @ B ) ) ) ).

% Fract_less_one_iff
thf(fact_9821_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_9822_rat__number__collapse_I5_J,axiom,
    ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% rat_number_collapse(5)
thf(fact_9823_Fract__add__one,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M @ N2 ) @ N2 )
        = ( plus_plus_rat @ ( fract @ M @ N2 ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_9824_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_9825_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% Fract_le_one_iff
thf(fact_9826_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_9827_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_9828_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9829_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9830_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_9831_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_9832_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_9833_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X5: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X5 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9834_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_9835_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_9836_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_9837_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y5: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9838_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y5: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y5 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y5 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y5 ) @ ( times_times_nat @ B @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y5 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9839_suminf__eq__SUP__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I4 ) )
       => ( ( 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_9840_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9841_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_9842_gcd__nat_Opelims,axiom,
    ! [X2: nat,Xa2: nat,Y2: nat] :
      ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y2 = X2 ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y2
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9843_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9844_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9845_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_9846_root__def,axiom,
    ( root
    = ( ^ [N: nat,X: real] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9847_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_9848_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_9849_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9850_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_9851_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_9852_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_9853_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_9854_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_9855_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_9856_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_9857_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_9858_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_9859_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_9860_upto_Opelims,axiom,
    ! [X2: int,Xa2: int,Y2: list_int] :
      ( ( ( upto @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa2 )
               => ( Y2
                  = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
               => ( Y2 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9861_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9862_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9863_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9864_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_9865_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_9866_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_9867_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_9868_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_9869_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_9870_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_9871_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_9872_upto__code,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_9873_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_9874_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9875_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_9876_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_9877_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_9878_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_9879_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_9880_upto_Oelims,axiom,
    ! [X2: int,Xa2: int,Y2: list_int] :
      ( ( ( upto @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( ord_less_eq_int @ X2 @ Xa2 )
         => ( Y2
            = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
         => ( Y2 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9881_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_9882_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_9883_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_9884_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_9885_DERIV__even__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X2 @ 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 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9886_DERIV__real__root__generic,axiom,
    ! [N2: nat,X2: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X2 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X2 )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X2 @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( 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 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9887_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 )
             => ? [Y3: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9888_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 )
             => ? [Y3: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y3 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9889_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_9890_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 )
             => ? [Y3: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y3 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9891_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 )
             => ? [Y3: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9892_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_9893_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ 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 @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9894_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [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 @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9895_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ 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 @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9896_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [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 @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9897_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_9898_DERIV__isconst3,axiom,
    ! [A: real,B: real,X2: real,Y2: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y2 @ ( 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 @ X2 )
              = ( F @ Y2 ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9899_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9900_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ L2 @ 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 @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9901_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ L2 @ 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 @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9902_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9903_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 ) ) ) )
       => ? [Z4: real] :
            ( ( ord_less_real @ A @ Z4 )
            & ( ord_less_real @ Z4 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z4 ) ) ) ) ) ) ).

% MVT2
thf(fact_9904_DERIV__local__const,axiom,
    ! [F: real > real,L2: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D )
             => ( ( F @ X2 )
                = ( F @ Y5 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9905_DERIV__ln,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9906_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_9907_DERIV__local__min,axiom,
    ! [F: real > real,L2: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y5 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9908_DERIV__local__max,axiom,
    ! [F: real > real,L2: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X2 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9909_DERIV__ln__divide,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9910_DERIV__pow,axiom,
    ! [N2: nat,X2: real,S3: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X2 @ S3 ) ) ).

% DERIV_pow
thf(fact_9911_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X2: real,N2: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N2 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X2 ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9912_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z5: real] : ( powr_real @ Z5 @ R )
        @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9913_DERIV__log,axiom,
    ! [X2: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9914_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X2: real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R )
          @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X2 ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9915_DERIV__powr,axiom,
    ! [G: real > real,M: real,X2: real,F: real > real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G @ X2 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9916_DERIV__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9917_DERIV__series_H,axiom,
    ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N4: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( F @ X @ N4 )
          @ ( F4 @ X0 @ N4 )
          @ ( 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 )
             => ( ! [N4: nat,X3: real,Y5: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y5 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y5 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
                  @ ( suminf_real @ ( F4 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9918_DERIV__arctan,axiom,
    ! [X2: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9919_arsinh__real__has__field__derivative,axiom,
    ! [X2: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9920_DERIV__real__sqrt__generic,axiom,
    ! [X2: real,D4: real] :
      ( ( X2 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X2 @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9921_arcosh__real__has__field__derivative,axiom,
    ! [X2: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9922_artanh__real__has__field__derivative,axiom,
    ! [X2: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9923_DERIV__power__series_H,axiom,
    ! [R2: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X3 @ N ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
       => ( ( ord_less_real @ zero_zero_real @ R2 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] :
                ( suminf_real
                @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ ( suc @ N ) ) ) )
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9924_DERIV__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9925_DERIV__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9926_DERIV__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9927_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X2: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
          & ( ( F @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9928_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X2: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
            & ( ( F @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9929_DERIV__odd__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X2 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9930_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 )
         => ( ! [M5: nat,T4: real] :
                ( ( ( ord_less_nat @ M5 @ N2 )
                  & ( ord_less_eq_real @ H2 @ T4 )
                  & ( ord_less_eq_real @ T4 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
           => ? [T4: real] :
                ( ( ord_less_real @ H2 @ T4 )
                & ( ord_less_real @ T4 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9931_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 )
       => ( ! [M5: nat,T4: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T4 )
                & ( ord_less_eq_real @ T4 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ? [T4: real] :
              ( ( ord_less_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9932_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 )
         => ( ! [M5: nat,T4: real] :
                ( ( ( ord_less_nat @ M5 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T4 )
                  & ( ord_less_eq_real @ T4 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
           => ? [T4: real] :
                ( ( ord_less_real @ zero_zero_real @ T4 )
                & ( ord_less_real @ T4 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9933_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X2: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X2 != zero_zero_real )
         => ( ! [M5: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T4: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
                & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
                & ( ( F @ X2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9934_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X2: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M5: nat,T4: real] :
            ( ( ( ord_less_nat @ M5 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X2 ) )
            & ( ( F @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X2 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9935_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 )
       => ( ! [M5: nat,T4: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T4: real] :
                  ( ( ord_less_real @ A @ T4 )
                  & ( ord_less_real @ T4 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9936_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 )
       => ( ! [M5: nat,T4: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T4: real] :
                  ( ( ord_less_real @ C @ T4 )
                  & ( ord_less_real @ T4 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9937_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T4: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X2 )
               => ( ( ord_less_eq_real @ X2 @ B )
                 => ( ( X2 != C )
                   => ? [T4: real] :
                        ( ( ( ord_less_real @ X2 @ C )
                         => ( ( ord_less_real @ X2 @ T4 )
                            & ( ord_less_real @ T4 @ C ) ) )
                        & ( ~ ( ord_less_real @ X2 @ C )
                         => ( ( ord_less_real @ C @ T4 )
                            & ( ord_less_real @ T4 @ X2 ) ) )
                        & ( ( F @ X2 )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ M6 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T4 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9938_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
      ( ! [M5: nat,T4: real] :
          ( ( ( ord_less_nat @ M5 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M2: nat,T6: real] :
            ( ( ( ord_less_nat @ M2 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U2 @ P4 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M2 ) ) )
                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T6 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T6 @ P4 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9939_DERIV__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X2 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9940_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_9941_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_9942_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_9943_take__bit__num__simps_I5_J,axiom,
    ! [R: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9944_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_9945_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_9946_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9947_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9948_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
        @ ^ [N: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N @ M ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9949_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ N2 @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] : ( some_num @ one )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9950_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
        @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9951_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M6: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9952_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_9953_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_9954_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_9955_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_9956_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9957_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_9958_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_9959_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_9960_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_9961_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9962_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_9963_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_9964_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_9965_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_9966_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M6: num] :
          ( produc478579273971653890on_num
          @ ^ [A4: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P4: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P4 ) )
                  @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
                  @ X )
              @ A4 )
          @ ( product_Pair_nat_num @ N @ M6 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9967_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,M8: real] :
            ( ! [X4: real] :
                ( ( ( ord_less_eq_real @ A @ X4 )
                  & ( ord_less_eq_real @ X4 @ B ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X4 ) )
                  & ( ord_less_eq_real @ ( F @ X4 ) @ M8 ) ) )
            & ! [Y3: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y3 )
                  & ( ord_less_eq_real @ Y3 @ M8 ) )
               => ? [X3: real] :
                    ( ( ord_less_eq_real @ A @ X3 )
                    & ( ord_less_eq_real @ X3 @ B )
                    & ( ( F @ X3 )
                      = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9968_isCont__real__sqrt,axiom,
    ! [X2: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9969_isCont__real__root,axiom,
    ! [X2: real,N2: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ ( root @ N2 ) ) ).

% isCont_real_root
thf(fact_9970_isCont__inverse__function2,axiom,
    ! [A: real,X2: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X2 )
     => ( ( ord_less_real @ X2 @ B )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B )
               => ( ( G @ ( F @ Z4 ) )
                  = Z4 ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_eq_real @ A @ Z4 )
               => ( ( ord_less_eq_real @ Z4 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9971_isCont__arcosh,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9972_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X2: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X2 )
         => ( ( ord_less_real @ X2 @ B )
           => ( ! [Y5: real] :
                  ( ( ord_less_real @ A @ Y5 )
                 => ( ( ord_less_real @ Y5 @ B )
                   => ( ( F @ ( G @ Y5 ) )
                      = Y5 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9973_isCont__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9974_isCont__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9975_LIM__less__bound,axiom,
    ! [B: real,X2: real,F: real > real] :
      ( ( ord_less_real @ B @ X2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9976_isCont__artanh,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9977_isCont__inverse__function,axiom,
    ! [D: real,X2: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
           => ( ( G @ ( F @ Z4 ) )
              = Z4 ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9978_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ A @ Z4 )
           => ( ( ord_less_eq_real @ Z4 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
               => ( ( ord_less_real @ Z4 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
           => ( ! [Z4: real] :
                  ( ( ord_less_real @ A @ Z4 )
                 => ( ( ord_less_real @ Z4 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9979_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9980_LIM__fun__not__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L2 != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ( F @ X4 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9981_LIM__fun__less__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9982_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
    @ ( 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_9983_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_9984_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_9985_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9986_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X: nat] : ( times_times_nat @ X @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9987_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_9988_monoseq__convergent,axiom,
    ! [X8: nat > real,B2: real] :
      ( ( topolo6980174941875973593q_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I4 ) ) @ B2 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_9989_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( root @ N @ ( semiri5074537144036343181t_real @ N ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9990_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
         => ( ( filterlim_nat_real
              @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
              @ ( 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_9991_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N7 @ N4 )
         => ( ord_less_real @ R3 @ ( X8 @ N4 ) ) )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( X8 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9992_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9993_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( root @ N @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9994_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9995_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9996_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L2: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N9: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9997_LIMSEQ__realpow__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_9998_LIMSEQ__divide__realpow__zero,axiom,
    ! [X2: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X2 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9999_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_10000_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_10001_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10002_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10003_tendsto__exp__limit__sequentially,axiom,
    ! [X2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_10004_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10005_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
          @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10006_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).

% summable
thf(fact_10007_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 )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10008_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10009_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
          @ ^ [N: 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 ) ) @ N ) ) )
          @ ( 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_10010_zeroseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10011_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( filterlim_nat_real
            @ ^ [N: 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 ) ) @ N ) ) )
            @ ( 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_10012_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( 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_10013_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ( 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
                @ ^ [N: 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 ) ) @ N ) ) )
                @ ( 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
                @ ^ [N: 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 ) ) @ N ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10014_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
          @ ^ [N: 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 ) ) @ N ) @ 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_10015_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( 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_10016_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
         => ( filterlim_nat_real
            @ ^ [N: 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 ) ) @ N ) @ 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_10017_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F3: real > real] :
        ? [C2: real] :
          ( F3
          = ( ^ [X: real] : ( times_times_real @ X @ C2 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10018_tendsto__exp__limit__at__right,axiom,
    ! [X2: real] :
      ( filterlim_real_real
      @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X2 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10019_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_10020_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_10021_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10022_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10023_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_10024_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_10025_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_10026_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_10027_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B )
         => ? [Y3: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y3 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10028_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
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
            @ at_bot_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10029_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_10030_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
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
            @ at_top_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10031_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_10032_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_10033_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_10034_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ X )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_10035_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10036_tendsto__exp__limit__at__top,axiom,
    ! [X2: real] :
      ( filterlim_real_real
      @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ Y ) ) @ Y )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10037_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_10038_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B @ X3 )
         => ? [Y3: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y3 @ 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_10039_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_10040_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ X2 )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ X2 )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_10041_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_10042_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10043_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_10044_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N6: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N6 @ N )
           => ( P @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_10045_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_10046_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_10047_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_10048_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X: real] : ( P @ ( plus_plus_real @ X @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_10049_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_10050_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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_10051_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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_10052_lhopital,axiom,
    ! [F: real > real,X2: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10053_lhopital__left,axiom,
    ! [F: real > real,X2: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10054_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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_10055_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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_10056_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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_10057_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ X2 )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ X2 )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10058_lhopital__at__top,axiom,
    ! [G: real > real,X2: real,G2: real > real,F: real > real,F4: real > real,Y2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10059_lhopital__left__at__top,axiom,
    ! [G: real > real,X2: real,G2: real > real,F: real > real,F4: real > real,Y2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10060_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
            @ ^ [X: real] :
                ( ( G0 @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F0 @ X ) @ ( G0 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_10061_lhopital__right,axiom,
    ! [F: real > real,X2: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F5
                    @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_10062_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
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_10063_lhopital__right__at__top,axiom,
    ! [G: real > real,X2: real,G2: real > real,F: real > real,F4: real > real,Y2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F4 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5849166863359141190n_real @ X2 ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_10064_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_10065_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_10066_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_10067_Bseq__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X2 ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10068_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
        = Y2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y2
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y2
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( 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
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( 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 ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_10069_decseq__bounded,axiom,
    ! [X8: nat > real,B2: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ B2 @ ( X8 @ I4 ) )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10070_decseq__convergent,axiom,
    ! [X8: nat > real,B2: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ B2 @ ( X8 @ I4 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I2: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I2 ) ) ) ) ) ).

% decseq_convergent
thf(fact_10071_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X @ ( 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 @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X5 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          @ ( 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 ) ) ) ) ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_10072_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( 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
                  @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                  @ ( 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 ) ) ) ) ) )
                           => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X5 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_10073_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_VEBT_valid @ X4 @ ( 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
                    @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                    @ ( 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 ) ) ) ) ) )
                             => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X5 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_10074_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( 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] :
                ( ( X2
                  = ( 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
                      @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                      @ ( 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 ) ) ) ) ) )
                               => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X5 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_10075_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( 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] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( 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
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( 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 ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_10076_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y2
                  = ( 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] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y2
                    = ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( 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
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( 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 ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ 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_10077_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,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ 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_10078_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,Z4: real] :
              ( ( ord_less_real @ A @ Z4 )
              & ( ord_less_real @ Z4 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10079_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_10080_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
          @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10081_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 )
       => ? [C3: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
              = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
            & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10082_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_10083_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_10084_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_10085_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
          @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_10086_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 ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
                & ( ord_less_real @ Z4 @ B )
                & ( ( F4 @ Z4 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10087_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_10088_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 )
             => ? [Y3: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y3 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10089_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 )
             => ? [Y3: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y3 @ 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_10090_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_10091_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X2: 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 @ X2 )
           => ( ( ord_less_eq_real @ X2 @ B )
             => ( ( F @ X2 )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10092_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 ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
                & ( ord_less_real @ Z4 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10093_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_10094_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10095_incseq__bounded,axiom,
    ! [X8: nat > real,B2: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B2 )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_10096_incseq__convergent,axiom,
    ! [X8: nat > real,B2: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B2 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I2: nat] : ( ord_less_eq_real @ ( X8 @ I2 ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_10097_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10098_and__not__num_Oelims,axiom,
    ! [X2: num,Xa2: num,Y2: option_num] :
      ( ( ( bit_and_not_num @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( X2 = one )
         => ( ( Xa2 = one )
           => ( Y2 != none_num ) ) )
       => ( ( ( X2 = one )
           => ( ? [N4: num] :
                  ( Xa2
                  = ( bit0 @ N4 ) )
             => ( Y2
               != ( some_num @ one ) ) ) )
         => ( ( ( X2 = one )
             => ( ? [N4: num] :
                    ( Xa2
                    = ( bit1 @ N4 ) )
               => ( Y2 != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X2
                    = ( bit0 @ M5 ) )
                 => ( ( Xa2 = one )
                   => ( Y2
                     != ( some_num @ ( bit0 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X2
                      = ( bit0 @ M5 ) )
                   => ! [N4: num] :
                        ( ( Xa2
                          = ( bit0 @ N4 ) )
                       => ( Y2
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N4 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X2
                        = ( bit0 @ M5 ) )
                     => ! [N4: num] :
                          ( ( Xa2
                            = ( bit1 @ N4 ) )
                         => ( Y2
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N4 ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X2
                          = ( bit1 @ M5 ) )
                       => ( ( Xa2 = one )
                         => ( Y2
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X2
                            = ( bit1 @ M5 ) )
                         => ! [N4: num] :
                              ( ( Xa2
                                = ( bit0 @ N4 ) )
                             => ( Y2
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M5 @ N4 ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X2
                              = ( bit1 @ M5 ) )
                           => ! [N4: num] :
                                ( ( Xa2
                                  = ( bit1 @ N4 ) )
                               => ( Y2
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_10099_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
              @ ^ [X: nat] : ( F @ ( G @ X ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_10100_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_10101_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_10102_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_10103_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_10104_and__num_Oelims,axiom,
    ! [X2: num,Xa2: num,Y2: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( X2 = one )
         => ( ( Xa2 = one )
           => ( Y2
             != ( some_num @ one ) ) ) )
       => ( ( ( X2 = one )
           => ( ? [N4: num] :
                  ( Xa2
                  = ( bit0 @ N4 ) )
             => ( Y2 != none_num ) ) )
         => ( ( ( X2 = one )
             => ( ? [N4: num] :
                    ( Xa2
                    = ( bit1 @ N4 ) )
               => ( Y2
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M5: num] :
                    ( X2
                    = ( bit0 @ M5 ) )
               => ( ( Xa2 = one )
                 => ( Y2 != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X2
                      = ( bit0 @ M5 ) )
                   => ! [N4: num] :
                        ( ( Xa2
                          = ( bit0 @ N4 ) )
                       => ( Y2
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N4 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X2
                        = ( bit0 @ M5 ) )
                     => ! [N4: num] :
                          ( ( Xa2
                            = ( bit1 @ N4 ) )
                         => ( Y2
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N4 ) ) ) ) )
                 => ( ( ? [M5: num] :
                          ( X2
                          = ( bit1 @ M5 ) )
                     => ( ( Xa2 = one )
                       => ( Y2
                         != ( some_num @ one ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X2
                            = ( bit1 @ M5 ) )
                         => ! [N4: num] :
                              ( ( Xa2
                                = ( bit0 @ N4 ) )
                             => ( Y2
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N4 ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X2
                              = ( bit1 @ M5 ) )
                           => ! [N4: num] :
                                ( ( Xa2
                                  = ( bit1 @ N4 ) )
                               => ( Y2
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M5 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_10105_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_10106_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_10107_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_10108_and__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_10109_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_10110_and__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_10111_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_10112_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_10113_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_10114_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_10115_xor__num_Oelims,axiom,
    ! [X2: num,Xa2: num,Y2: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X2 @ Xa2 )
        = Y2 )
     => ( ( ( X2 = one )
         => ( ( Xa2 = one )
           => ( Y2 != none_num ) ) )
       => ( ( ( X2 = one )
           => ! [N4: num] :
                ( ( Xa2
                  = ( bit0 @ N4 ) )
               => ( Y2
                 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
         => ( ( ( X2 = one )
             => ! [N4: num] :
                  ( ( Xa2
                    = ( bit1 @ N4 ) )
                 => ( Y2
                   != ( some_num @ ( bit0 @ N4 ) ) ) ) )
           => ( ! [M5: num] :
                  ( ( X2
                    = ( bit0 @ M5 ) )
                 => ( ( Xa2 = one )
                   => ( Y2
                     != ( some_num @ ( bit1 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X2
                      = ( bit0 @ M5 ) )
                   => ! [N4: num] :
                        ( ( Xa2
                          = ( bit0 @ N4 ) )
                       => ( Y2
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N4 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X2
                        = ( bit0 @ M5 ) )
                     => ! [N4: num] :
                          ( ( Xa2
                            = ( bit1 @ N4 ) )
                         => ( Y2
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N4 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X2
                          = ( bit1 @ M5 ) )
                       => ( ( Xa2 = one )
                         => ( Y2
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X2
                            = ( bit1 @ M5 ) )
                         => ! [N4: num] :
                              ( ( Xa2
                                = ( bit0 @ N4 ) )
                             => ( Y2
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N4 ) ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X2
                              = ( bit1 @ M5 ) )
                           => ! [N4: num] :
                                ( ( Xa2
                                  = ( bit1 @ N4 ) )
                               => ( Y2
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_10116_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_10117_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_10118_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_10119_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_10120_xor__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ ( bit1 @ N2 ) ) ) ).

% xor_num.simps(2)
thf(fact_10121_xor__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ ( bit0 @ N2 ) ) ) ).

% xor_num.simps(3)
thf(fact_10122_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_10123_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_10124_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_10125_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_10126_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_10127_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10128_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_10129_inj__Suc,axiom,
    ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).

% inj_Suc
thf(fact_10130_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member_nat @ N4 @ N3 )
         => ( ord_less_eq_nat @ K @ N4 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N3 ) ) ).

% inj_on_diff_nat
thf(fact_10131_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_10132_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_10133_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_10134_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_10135_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_10136_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_10137_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_10138_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_10139_powr__real__of__int_H,axiom,
    ! [X2: real,N2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( X2 != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N2 ) )
       => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N2 ) )
          = ( power_int_real @ X2 @ N2 ) ) ) ) ).

% powr_real_of_int'
thf(fact_10140_remdups__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( remdups_nat @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% remdups_upt
thf(fact_10141_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_10142_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10143_sorted__list__of__set__range,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sorted_list_of_set_range
thf(fact_10144_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_10145_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_10146_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10147_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_10148_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M6 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10149_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M6 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10150_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N: nat,M6: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M6 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10151_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10152_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_10153_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_10154_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_10155_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_10156_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_10157_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_10158_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_10159_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_10160_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10161_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X2: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X2 @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X2 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10162_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10163_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_10164_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_10165_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).

% sum_list_upt
thf(fact_10166_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L: list_nat] :
                ( ( ( size_size_list_nat @ L )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L )
                  = N3 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L: list_nat] :
                  ( ( ( size_size_list_nat @ L )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L )
                    = N3 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L: list_nat] :
                  ( ( ( size_size_list_nat @ L )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
                    = N3 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_10167_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L: list_nat] :
              ( ( ( size_size_list_nat @ L )
                = M )
              & ( ( groups4561878855575611511st_nat @ L )
                = N3 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).

% card_length_sum_list
thf(fact_10168_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_10169_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_10170_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_10171_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_10172_sorted__upto,axiom,
    ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).

% sorted_upto
thf(fact_10173_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_10174_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_10175_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_10176_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_10177_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] :
          ! [M6: nat] :
            ( ( ord_less_eq_nat @ N6 @ M6 )
           => ! [N: nat] :
                ( ( ord_less_eq_nat @ N6 @ N )
               => ( P @ ( product_Pair_nat_nat @ N @ M6 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10178_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ M ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
        @ ^ [R4: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R4 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_10179_at__right__to__0,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
      = ( filtermap_real_real
        @ ^ [X: real] : ( plus_plus_real @ X @ A )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_right_to_0
thf(fact_10180_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_10181_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_10182_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_10183_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_10184_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% min_numeral_Suc
thf(fact_10185_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_10186_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_10187_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_10188_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).

% min_diff
thf(fact_10189_concat__bit__assoc__sym,axiom,
    ! [M: nat,N2: nat,K: int,L2: int,R: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N2 @ K @ L2 ) @ R )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N2 ) @ L2 @ R ) ) ) ).

% concat_bit_assoc_sym
thf(fact_10190_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N2: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N2 @ K @ L2 ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ L2 ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_10191_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M3: nat] : ( suc @ ( ord_min_nat @ N2 @ M3 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_10192_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M3: nat] : ( suc @ ( ord_min_nat @ M3 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_10193_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M6: nat,N: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ) ).

% pred_nat_def
thf(fact_10194_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_10195_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_10196_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% less_eq
thf(fact_10197_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_10198_take__upt,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
     => ( ( take_nat @ M @ ( upt @ I @ N2 ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_10199_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
thf(fact_10200_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_10201_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,N: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N ) ) )
          & ( N != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_10202_Rats__abs__iff,axiom,
    ! [X2: real] :
      ( ( member_real @ ( abs_abs_real @ X2 ) @ field_5140801741446780682s_real )
      = ( member_real @ X2 @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_10203_Rats__no__bot__less,axiom,
    ! [X2: real] :
    ? [X3: real] :
      ( ( member_real @ X3 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X3 @ X2 ) ) ).

% Rats_no_bot_less
thf(fact_10204_Rats__dense__in__real,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ? [X3: real] :
          ( ( member_real @ X3 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X2 @ X3 )
          & ( ord_less_real @ X3 @ Y2 ) ) ) ).

% Rats_dense_in_real
thf(fact_10205_Rats__no__top__le,axiom,
    ! [X2: real] :
    ? [X3: real] :
      ( ( member_real @ X3 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X2 @ X3 ) ) ).

% Rats_no_top_le
thf(fact_10206_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,J3: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_10207_finite__vimage__Suc__iff,axiom,
    ! [F5: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F5 ) )
      = ( finite_finite_nat @ F5 ) ) ).

% finite_vimage_Suc_iff
thf(fact_10208_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
      = ( vimage_nat_nat @ suc @ A2 ) ) ).

% vimage_Suc_insert_0
thf(fact_10209_vimage__Suc__insert__Suc,axiom,
    ! [N2: nat,A2: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N2 ) @ A2 ) )
      = ( insert_nat @ N2 @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_10210_set__decode__div__2,axiom,
    ! [X2: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X2 ) ) ) ).

% set_decode_div_2
thf(fact_10211_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc

% Helper facts (40)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y2: int] :
      ( ( if_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y2: int] :
      ( ( if_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X2: num,Y2: num] :
      ( ( if_num @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X2: num,Y2: num] :
      ( ( if_num @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( if_rat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( if_rat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y2: real] :
      ( ( if_real @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y2: real] :
      ( ( if_real @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X5: real] : ( P @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( if_complex @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( if_complex @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y2: extended_enat] :
      ( ( if_Extended_enat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y2: extended_enat] :
      ( ( if_Extended_enat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( if_Code_integer @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( if_Code_integer @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( if_set_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( if_set_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( if_set_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( if_set_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( if_list_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( if_list_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y2: option_nat] :
      ( ( if_option_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y2: option_nat] :
      ( ( if_option_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X2: option_num,Y2: option_num] :
      ( ( if_option_num @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X2: option_num,Y2: option_num] :
      ( ( if_option_num @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y2: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y2: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X2: produc6271795597528267376eger_o,Y2: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X2: produc6271795597528267376eger_o,Y2: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y2 )
      = X2 ) ).

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,
    ! [X2: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( ord_less_nat
      @ ( if_nat
        @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
          = ma )
        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
        @ ma )
      @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) )
    & ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
      @ ( if_nat
        @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
          = ma )
        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
        @ ma ) ) ) ).

%------------------------------------------------------------------------------